Recent zbMATH articles in MSC 65https://www.zbmath.org/atom/cc/652021-02-27T13:50:00+00:00WerkzeugCahn-Hilliard vs singular Cahn-Hilliard equations in simulations of immiscible binary fluids.https://www.zbmath.org/1453.652122021-02-27T13:50:00+00:00"Chen, Lizhen"https://www.zbmath.org/authors/?q=ai:chen.lizhenSummary: An efficient semi-implicit spectral method is implemented to solve the Cahn-Hilliard equation with a variable mobility in this paper. We compared the kinetics of bulk-diffusion-dominated and interface-diffusion-dominated coarsening in two-phase systems. As expected, the interface-diffusion-controlled coarsening evolves much slower. Also we find that the velocity field will be caused different greatly by using Singular Cahn-Hilliard equation and using Cahn-Hilliard in the simulation of immiscible binary fluids.Choice of finite-difference schemes in solving coefficient inverse problems.https://www.zbmath.org/1453.652882021-02-27T13:50:00+00:00"Albu, A. F."https://www.zbmath.org/authors/?q=ai:albu.alla-f"Evtushenko, Yu. G."https://www.zbmath.org/authors/?q=ai:evtushenko.yuri-g"Zubov, V. I."https://www.zbmath.org/authors/?q=ai:zubov.vladimir-ivanovich.1Summary: Various choices of a finite-difference scheme for approximating the heat diffusion equation in solving a three-dimensional coefficient inverse problem were studied. A comparative analysis was conducted for several alternating direction schemes, such as locally one-dimensional, Douglas-Rachford, and Peaceman-Rachford schemes, as applied to nonlinear problems for the three-dimensional heat equation with temperature-dependent coefficients. Each numerical method was used to compute the temperature distribution inside a parallelepiped. The methods were compared in terms of the accuracy of the resulting solution and the computation time required for achieving the prescribed accuracy on a computer.Refactorization of the midpoint rule.https://www.zbmath.org/1453.651572021-02-27T13:50:00+00:00"Burkardt, John"https://www.zbmath.org/authors/?q=ai:burkardt.john-v"Trenchea, Catalin"https://www.zbmath.org/authors/?q=ai:trenchea.catalinSummary: An alternative formulation of the midpoint method is employed to analyze its advantages as an implicit second-order absolutely stable timestepping method. Legacy codes originally using the backward Euler method can be upgraded to this method by inserting a single line of new code. We show that the midpoint method, and a theta-like generalization, are B-stable. We outline two estimates of local truncation error that allow adaptive time-stepping.Range-separated tensor decomposition of the discretized Dirac delta and elliptic operator inverse.https://www.zbmath.org/1453.650352021-02-27T13:50:00+00:00"Khoromskij, Boris N."https://www.zbmath.org/authors/?q=ai:khoromskij.boris-nSummary: In this paper, we introduce the operator dependent range-separated (RS) tensor approximation of the discretized Dirac delta function (distribution) in \(\mathbb{R}^d\). It is constructed by application of the elliptic operator to the RS tensor representation of the associated Green kernel discretized on the \(d\)-dimensional Cartesian grid. The proposed local-global decomposition of the Dirac delta can be applied for solving the potential equations in a non-homogeneous medium when the density in the right-hand side is given by a large sum of pointwise singular charges. As an example of applications, we describe the regularization scheme for solving the Poisson-Boltzmann equation that models the electrostatics in bio-molecules. We show how the idea of the operator dependent RS tensor decomposition of the Dirac delta can be generalized to the closely related problem on range-separated tensor representation of the elliptic resolvent. This approach paves the way for application of tensor numerical methods to elliptic problems with non-regular data. Numerical tests confirm the expected localization properties of the RS tensor approximation of the Dirac delta represented on a tensor grid in 3D.A uniformly convergent finite difference scheme for Robin type singularly perturbed parabolic convection diffusion problem.https://www.zbmath.org/1453.652292021-02-27T13:50:00+00:00"Mbroh, Nana Adjoah"https://www.zbmath.org/authors/?q=ai:mbroh.nana-adjoah"Noutchie, Suares Clovis Oukouomi"https://www.zbmath.org/authors/?q=ai:oukouomi-noutchie.suares-clovis"Massoukou, Rodrigue Yves M'pika"https://www.zbmath.org/authors/?q=ai:massoukou.rodrigue-yves-mpikaSummary: In this paper, a second order numerical scheme for solving a singularly perturbed convection diffusion problem with Robin boundary conditions is proposed. The numerical scheme is a combination of the fitted operator finite difference and the backward Euler finite difference methods. These are designed in order to solve respectively the spatial derivatives and the time derivative. Using some properties of the discrete problem the methods are analysed for convergence. Richardson extrapolation technique is used to improve the accuracy and also accelerate the convergence of the method. Numerical simulations are carried out to confirm the theoretical findings in the analysis before and after extrapolation.A high order method for pricing of financial derivatives using radial basis function generated finite differences.https://www.zbmath.org/1453.911082021-02-27T13:50:00+00:00"Milovanović, Slobodan"https://www.zbmath.org/authors/?q=ai:milovanovic.slobodan"von Sydow, Lina"https://www.zbmath.org/authors/?q=ai:von-sydow.linaSummary: In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not requiring Cartesian grids. Instead, the nodes can be placed with higher density in areas where there is a need for higher accuracy. Still, the discretization matrix is fairly sparse. As a model problem, we consider the pricing of European options in 2D. Since such options have a discontinuity in the first derivative of the payoff function which prohibits high order convergence, we smooth this function using an established technique for Cartesian grids. Numerical experiments show that we acquire a fourth order scheme in space, both for the uniform and the nonuniform node layouts that we use. The high order method with the nonuniform node layout achieves very high accuracy with relatively few nodes. This renders the potential for solving pricing problems in higher spatial dimensions since the computational memory and time demand become much smaller with this method compared to standard techniques.Dynamic large deformation analysis of a cantilever beam.https://www.zbmath.org/1453.740522021-02-27T13:50:00+00:00"Wei, H."https://www.zbmath.org/authors/?q=ai:wei.hongbo|wei.hongxu|wei.hua|wei.hu|wei.haixin|wei.hongkai|wei.hai-rui|wei.he|wei.huanming|wei.huiqi|wei.hongji|wei.hongchuan|wei.huaying|wei.hongning|wei.hongjun|wei.hui|wei.huailiang|wei.huifang|wei.huachun|wei.huanmin|wei.hanbing|wei.henghua|wei.honggui|wei.heng|wei.hao|wei.huan|wei.haihua|wei.hongyan|wei.hsienhung|wei.hongchang|wei.huihui|wei.hewen|wei.hongxing|wei.huayi|wei.hongjie|wei.haigang|wei.hongzhi|wei.hezhong|wei.huaquan|wei.heqing|wei.hanbin|wei.hongxia|wei.haoyuan|wei.haiguang|wei.hongduo|wei.haiyang|wei.hongzhen|wei.hong|wei.hongru|wei.hongyun|wei.hongjin|wei.huang|wei.hongzeng|wei.haiying|wei.haie|wei.hanbai|wei.huiming|wei.hejie|wei.hanying|wei.hengjia|wei.hengdong|wei.hang|wei.hanlin|wei.haijun|wei.haili|wei.hengfeng|wei.haoyan|wei.huiwen|wei.hongbin|wei.hualiang|wei.honglei|wei.haizhou|wei.hanyu|wei.han|wei.hengyang|wei.huixian|wei.haikun|wei.hongyu"Pan, Q. X."https://www.zbmath.org/authors/?q=ai:pan.qingxian|pan.qinxue|pan.quanxiang|pan.qunxing"Adetoro, O. B."https://www.zbmath.org/authors/?q=ai:adetoro.o-b"Avital, E."https://www.zbmath.org/authors/?q=ai:avital.eldad-j"Yuan, Y."https://www.zbmath.org/authors/?q=ai:yuan.yali|yuan.yongfeng|yuan.yufei|yuan.yipu|yuan.yueyun|yuan.yubo|yuan.yudong|yuan.yingchun|yuan.yandong|yuan.yuxiang|yuan.yingzhong|yuan.yunyao|yuan.yongjun|yuan.yao|yuan.youguang|yuan.yiping|yuan.yujun|yuan.yaping|yuan.yuze|yuan.yiran|yuan.yanan|yuan.yefei|yuan.ying|yuan.yujie|yuan.yiwu|yuan.yougaung|yuan.yongming|yuan.yabo|yuan.yahua|yuan.yinglong|yuan.yanhong|yuan.yuling|yuan.yanbin|yuan.ying.1|yuan.yanbo|yuan.yige|yuan.yangsheng|yuan.yuzhuo|yuan.yayan|yuan.yong|yuan.yilin|yuan.yifan|yuan.yilian|yuan.yu|yuan.yuehua|yuan.yanxiang|yuan.yun|yuan.yingying|yuan.yifei|yuan.yiwei|yuan.ya-xiang|yuan.yinghe|yuan.yueding|yuan.yongjiu|yuan.yanhui|yuan.yu.1|yuan.yanchao|yuan.yinghai|yuan.yingna|yuan.yunyue|yuan.youyou|yuan.yuemei|yuan.yuchen|yuan.yule|yuan.yifeng|yuan.yang|yuan.yanfei|yuan.yueming|yuan.yeli|yuan.yunhe|yuan.yuping|yuan.yongke|yuan.yixing|yuan.yutang|yuan.yancheng|yuan.yan|yuan.yunliang|yuan.yuming|yuan.yueshuang|yuan.yi|yuan.yongbin|yuan.yunbin|yuan.yinzhong|yuan.yuan|yuan.yuejin|yuan.yuanlong|yuan.yuan.2|yuan.yuhao|yuan.yuan.3|yuan.yongsheng|yuan.yunfei|yuan.yanyan|yuan.youcheng|yuan.yongxing|yuan.yizheng|yuan.yupeng|yuan.yahong|yuan.yanpeng|yuan.yijia|yuan.yuan.1|yuan.yue|yuan.ye|yuan.yantao|yuan.youwei|yuan.yuhai|yuan.yongna|yuan.yuhuan|yuan.yongzhuang|yuan.yujing|yuan.yanli|yuan.yanhua|yuan.yumin|yuan.yinlong|yuan.yongxin|yuan.yirang"Wen, P. H."https://www.zbmath.org/authors/?q=ai:wen.peihan|wen.pihua-h|wen.pihuaSummary: A static and dynamic large deformation analysis of a tapered beam subjected to concentrated and distributed loads is presented in this paper by using a direct integration technique. The bending stiffness of the beam is coordinate dependent. The nonlinear differential equation is numerically solved using an iterative technique without an algebraic equation solver, thus the computational effort can be reduced. A concentrated mass fixed at the free end and suddenly released is studied, and the time-dependent displacements are presented. Comparison has been made with solutions obtained using Finite Element Analysis and excellent agreement is achieved.A QSC method for fractional subdiffusion equations with fractional boundary conditions and its application in parameters identification.https://www.zbmath.org/1453.653592021-02-27T13:50:00+00:00"Liu, Jun"https://www.zbmath.org/authors/?q=ai:liu.jun.1|liu.jun.3|liu.jun.2|liu.jun.5|liu.jun.4|liu.jun"Fu, Hongfei"https://www.zbmath.org/authors/?q=ai:fu.hongfei"Zhang, Jiansong"https://www.zbmath.org/authors/?q=ai:zhang.jiansongSummary: A quadratic spline collocation (QSC) method combined with \(L 1\) time discretization, named QSC-\( L 1\), is proposed to solve fractional subdiffusion equations with artificial boundary conditions. A novel norm-based stability and convergence analysis is carefully discussed, which shows that the QSC-\( L 1\) method is unconditionally stable in a discrete space-time norm, and has a convergence order \(\mathcal{O} (\tau^{2-\alpha} + h^2)\), where \(\tau\) and \(h\) are the temporal and spatial step sizes, respectively. Then, based on fast evaluation of the Caputo fractional derivative (see, [\textit{S. Jiang} et al., ``Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations'', Commun. Comput. Phys. 21, No. 3, 650--678 (2017; \url{doi:10.4208/cicp.oa-2016-0136})]), a fast version of QSC-\(L1\) which is called QSC-F\(L1\) is proposed to improve the computational efficiency. Two numerical examples are provided to support the theoretical results. Furthermore, an inverse problem is considered, in which some parameters of the fractional subdiffusion equations need to be identified. A Levenberg-Marquardt (L-M) method equipped with the QSC-F\(L1\) method is developed for solving the inverse problem. Numerical tests show the effectiveness of the method even for the case that the observation data is contaminated by some levels of random noise.Haar wavelet quasilinearization method for numerical solution of Emden-Fowler type equations.https://www.zbmath.org/1453.651962021-02-27T13:50:00+00:00"Singh, Randhir"https://www.zbmath.org/authors/?q=ai:singh.randhir"Guleria, Vandana"https://www.zbmath.org/authors/?q=ai:guleria.vandana"Singh, Mehakpreet"https://www.zbmath.org/authors/?q=ai:singh.mehakpreetSummary: In this paper, an efficient method for solving the nonlinear Emden-Fowler type boundary value problems with Dirichlet and Robin-Neumann boundary conditions is introduced. The present method is based on the Haar-wavelets and quasilinearization technique. The quasilinearization technique is adopted to linearize the nonlinear singular problem. Numerical solution of linear singular problem is obtained by the Haar wavelet method. The numerical study is further supported by examining thoroughly the convergence of the Haar wavelet method and the quasilinearization technique. In order to check the accuracy of the proposed method, the numerical results are compared with both existing methods and exact solutions.High-order orthogonal spline collocation methods for two-point boundary value problems with interfaces.https://www.zbmath.org/1453.651902021-02-27T13:50:00+00:00"Bhal, Santosh Kumar"https://www.zbmath.org/authors/?q=ai:bhal.santosh-kumar"Danumjaya, P."https://www.zbmath.org/authors/?q=ai:danumjaya.palla"Fairweather, G."https://www.zbmath.org/authors/?q=ai:fairweather.graemeSummary: Orthogonal spline collocation methods (OSC) are used to solve two-point boundary value problems (BVPs) with interfaces. We first consider the one-dimensional Helmholtz equation with piecewise wave numbers solved using the standard OSC approach. For the solution of self-adjoint two-point BVPs with interfaces, we employ OSC with monomial bases of degree \(r\), where \(r = 3 , 4\). In each case, the results of numerical experiments involving numerous examples from the literature exhibit optimal accuracy in the \(L^\infty\) and \(L^2\) norms of order \(r + 1\), and order \(r\) accuracy in the \(H^1\) norm. Moreover, superconvergence of order \(2 r - 2\) in the nodal error in the OSC approximation and also in its derivative when \(r = 4\) is observed. Each OSC approach gives rise to almost block diagonal linear systems which are solved using standard software.Improved Runge-Kutta-Chebyshev methods.https://www.zbmath.org/1453.651682021-02-27T13:50:00+00:00"Tang, Xiao"https://www.zbmath.org/authors/?q=ai:tang.xiao"Xiao, Aiguo"https://www.zbmath.org/authors/?q=ai:xiao.aiguoSummary: This study proposes a class of improved Runge-Kutta-Chebyshev (RKC) methods for the stiff systems arising from the spatial discretization of partial differential equations. We can obtain the improved first-order and second-order RKC methods by introducing an appropriate combination technique. The main advantage of our improved RKC methods is that the width of the stability domain along the imaginary axis is significantly increased while the length along the negative real axis has almost no reduction. This implies that our improved RKC methods can extend the application scope of the classical RKC methods. The results of five numerical examples (including the advection-diffusion-reaction equations with dominating advection) show that our improved RKC methods can perform very well.Cubic B-spline Galerkin method for numerical solution of the coupled nonlinear Schrödinger equation.https://www.zbmath.org/1453.653252021-02-27T13:50:00+00:00"Iqbal, Azhar"https://www.zbmath.org/authors/?q=ai:iqbal.azhar.1|iqbal.azhar"Abd Hamid, Nur Nadiah"https://www.zbmath.org/authors/?q=ai:abd-hamid.nur-nadiah"Md. Ismail, Ahmad Izani"https://www.zbmath.org/authors/?q=ai:ismail.ahmad-izani-md|ismail.ahmad-izani-mohamedSummary: In this paper, the Galerkin method, based on cubic B-spline function as the shape and weight functions is applied for the numerical solution of the one-dimensional coupled nonlinear Schrödinger equation. Numerical experiments involving single solitary wave, collision of two solitary waves and collision of three solitary waves are conducted. The obtained numerical results of the proposed scheme are compared with the analytical results and previously published numerical results. Two conserved quantities \(I_1\) and \(I_2\) are calculated for collision of two solitary waves and interaction of three solitary waves. The scheme provides accurate results which are in good agreement when compared to other numerical schemes. The order of convergence of the scheme is calculated. Moreover, the use of cubic B-spline Galerkin method produces smooth solutions without numerical smearing.A general preconditioning framework for coupled multiphysics problems with application to contact- and poro-mechanics.https://www.zbmath.org/1453.650652021-02-27T13:50:00+00:00"Ferronato, Massimiliano"https://www.zbmath.org/authors/?q=ai:ferronato.massimiliano"Franceschini, Andrea"https://www.zbmath.org/authors/?q=ai:franceschini.andrea"Janna, Carlo"https://www.zbmath.org/authors/?q=ai:janna.carlo"Castelletto, Nicola"https://www.zbmath.org/authors/?q=ai:castelletto.nicola"Tchelepi, Hamdi A."https://www.zbmath.org/authors/?q=ai:tchelepi.hamdi-aSummary: This work discusses a general approach for preconditioning the block Jacobian matrix arising from the discretization and linearization of coupled multiphysics problem. The objective is to provide a fully algebraic framework that can be employed as a starting point for the development of specialized algorithms exploiting unique features of the specific problem at hand. The basic idea relies on approximately computing an operator able to decouple the different processes, which can then be solved independently one from the other. In this work, the decoupling operator is computed by extending the theory of block sparse approximate inverses. The proposed approach is implemented for two multiphysics applications, namely the simulation of a coupled poromechanical system and the mechanics of fractured media. The numerical results obtained in experiments taken from real-world examples are used to analyze and discuss the properties of the preconditioner.Singularly perturbed convection-diffusion boundary value problems with two small parameters using nonpolynomial spline technique.https://www.zbmath.org/1453.651722021-02-27T13:50:00+00:00"Khandelwal, Pooja"https://www.zbmath.org/authors/?q=ai:khandelwal.pooja"Khan, Arshad"https://www.zbmath.org/authors/?q=ai:khan.arshad-ali|khan.arshad-alam|khan.arshad-m|khan.arshad-ahmadSummary: In this paper, a new nonpolynomial cubic spline method is developed for solving two-parameter singularly perturbed boundary value problems. Convergence analysis is briefly discussed. Numerical examples and computational results illustrate and guarantee a higher accuracy by this technique. Comparisons are made to confirm the reliability and accuracy of the proposed technique.Wavelet-based numerical techniques for 1D peristatic problems in infinite domain.https://www.zbmath.org/1453.740822021-02-27T13:50:00+00:00"Singh, Debabrata"https://www.zbmath.org/authors/?q=ai:singh.debabrata"Panja, Madan Mohan"https://www.zbmath.org/authors/?q=ai:panja.madan-mohanSummary: Peristatics is an important branch of continuum mechanics dealing with nonlocal effects in solid structures. Some 1D peristatic problems with four different micromodulus functions have been studied here. A wavelet-based collocation method has been developed to find multiscale approximate solutions of the peristatic problems that avoid cumbersome evaluation of singular type integrals present in analytical solution available in literature. The obtained results offer important insights into applications and simulations of peristatic models and seem to be useful for investigation in the field of peridynamics in one or higher dimensions.The method of fundamental solutions for computing interior transmission eigenvalues of inhomogeneous media.https://www.zbmath.org/1453.653992021-02-27T13:50:00+00:00"Pieronek, Lukas"https://www.zbmath.org/authors/?q=ai:pieronek.lukas"Kleefeld, Andreas"https://www.zbmath.org/authors/?q=ai:kleefeld.andreasSummary: The method of fundamental solutions is applied to the approximate computation of interior transmission eigenvalues for a special class of inhomogeneous media in two dimensions. We give a short approximation analysis accompanied with numerical results that clearly prove practical convenience of our alternative approach.
For the entire collection see [Zbl 1417.65006].Variational extrapolation of implicit schemes for general gradient flows.https://www.zbmath.org/1453.652772021-02-27T13:50:00+00:00"Zaitzeff, Alexander"https://www.zbmath.org/authors/?q=ai:zaitzeff.alexander"Esedoḡlu, Selim"https://www.zbmath.org/authors/?q=ai:esedoglu.selim"Garikipati, Krishna"https://www.zbmath.org/authors/?q=ai:garikipati.krishnaThe authors introduce a class of unconditionally energy-stable, high-order-accurate schemes for gradient flows in a very general setting. The new schemes have the following advantages
\begin{itemize}
\item[(1)] Complete generality. There is no assumption (e.g., convexity) on the energy \(E\) in (1.1) beyond sufficient differentiability.
\item[(2)] Unconditional energy stability.
\item[(3)] High-order (at least up to third-order) accuracy.
\item[(4)] Each time step requires a few standard minimizing-movements solves, equivalent to backward Euler substeps, or optimization of the associated energy plus a quadratic term.
\end{itemize}
This approach gives a painless way to extend these to high-order-accurate-in-time schemes while maintaining their unconditional stability. It also can be viewed as a variational analogue of Richardson extrapolation.
Reviewer: Qifeng Zhang (Hangzhou)A second-order stabilization method for linearizing and decoupling nonlinear parabolic systems.https://www.zbmath.org/1453.652262021-02-27T13:50:00+00:00"Li, Buyang"https://www.zbmath.org/authors/?q=ai:li.buyang"Ueda, Yuki"https://www.zbmath.org/authors/?q=ai:ueda.yuki"Zhou, Guanyu"https://www.zbmath.org/authors/?q=ai:zhou.guanyuThe authors construct a new time discretization method for strongly nonlinear parabolic systems by combining the fully explicit two-step backward difference formula and a second-order stabilization of wave type. The proposed method linearizes and decouples a nonlinear parabolic system at every time level, with second-order consistency error, which is accurate and time-saving. The convergence of the proposed method is proved by combining energy estimates for evolution equations of parabolic and wave types with the generating function technique that is popular in studying ordinary differential equations. Numerical examples are provided to support their theoretical result.
Reviewer: Qifeng Zhang (Hangzhou)The nodal \(LTS_N\) solution in a rectangular domain: a new method to determine the outgoing angular flux at the boundary.https://www.zbmath.org/1453.821002021-02-27T13:50:00+00:00"Parigi, Aline R."https://www.zbmath.org/authors/?q=ai:parigi.aline-r"Segatto, Cynthia F."https://www.zbmath.org/authors/?q=ai:segatto.cynthia-f"Bodmann, Bardo E. J."https://www.zbmath.org/authors/?q=ai:bodmann.bardo-ernst-josefSummary: In the present contribution we discuss the neutron nodal \(S_N\) equation in a rectangular domain. The nodal method consists in transverse integration of the \(S_N\) equation and results in coupled one-dimensional \(S_N\) equations with unknown angular flux at the border. In the literature, the outgoing angular flux is considered a constant or exponential decreasing function, where the latter is used in this work. It is noteworthy that solutions found with these boundary conditions present unphysical results, i.e. negative angular fluxes in the border region, whereas the scalar flux is semi-positive definite. To overcome these shortcomings a new approach is proposed. The rectangular domain is covered by a finite discrete set of narrow rectangular sub-domains, so that in each rectangle the solution may be approximated by the one from a one-dimensional problem. Upon applying the \(LTS_N\) method combined with the DNI technique, i.e. interpolating the directions of the two-dimensional problem by means of one-dimensional directions, one obtains the angular flux at the border from the known one-dimensional \(LTS_N\) solution for any desired point. Numerical simulations and comparisons with results found in the literature are presented.
For the entire collection see [Zbl 1417.65006].Error bounds of the finite difference time domain methods for the Dirac equation in the semiclassical regime.https://www.zbmath.org/1453.652282021-02-27T13:50:00+00:00"Ma, Ying"https://www.zbmath.org/authors/?q=ai:ma.ying"Yin, Jia"https://www.zbmath.org/authors/?q=ai:yin.jiaFour finite difference time domain (FDTD) methods, the leap-frog, two semi-implicit, and the Crank-Nicolson method, are applied to solve the Dirac equation numerically in the semiclassical regime \(0 < \varepsilon \le 1\), where \(\varepsilon\) is representing the scaled Planck constant. The stability conditions and error estimates of the FDTD methods suggest that all these FDTD methods share the same \(\varepsilon\)-scalability at \(\tau = {\mathcal O}(\varepsilon^{3/2})\) and \(h= {\mathcal O}(\varepsilon^{3/2})\), where \(\tau\) is the time step size and \(h\) the mesh size in space. The computational cost for the leap-frog method is the lowest, and that for the Crank-Nicolson method is the highest. The total probability density and current density also show the same \(\varepsilon\)-scalability. The error estimates are validated by extensive numerical experiments.
Reviewer: Bülent Karasözen (Ankara)A posteriori error estimates for the Allen-Cahn problem.https://www.zbmath.org/1453.653122021-02-27T13:50:00+00:00"Chrysafinos, Konstantinos"https://www.zbmath.org/authors/?q=ai:chrysafinos.konstantinos"Georgoulis, Emmanuil H."https://www.zbmath.org/authors/?q=ai:georgoulis.emmanuil-h"Plaka, Dimitra"https://www.zbmath.org/authors/?q=ai:plaka.dimitraMirror descent algorithms for minimizing interacting free energy.https://www.zbmath.org/1453.651452021-02-27T13:50:00+00:00"Ying, Lexing"https://www.zbmath.org/authors/?q=ai:ying.lexingSummary: This note considers the problem of minimizing interacting free energy. Motivated by the mirror descent algorithm, for a given interacting free energy, we propose a descent dynamics with a novel metric that takes into consideration the reference measure and the interacting term. This metric naturally suggests a monotone reparameterization of the probability measure. By discretizing the reparameterized descent dynamics with the explicit Euler method, we arrive at a new mirror-descent-type algorithm for minimizing interacting free energy. Numerical results are included to demonstrate the efficiency of the proposed algorithms.Recent advances in domain decomposition methods for total variation minimization.https://www.zbmath.org/1453.654292021-02-27T13:50:00+00:00"Lee, Chang-Ock"https://www.zbmath.org/authors/?q=ai:lee.chang-ock"Park, Jongho"https://www.zbmath.org/authors/?q=ai:park.jonghoSummary: Total variation minimization is standard in mathematical imaging and there have been numerous researches over the last decades. In order to process large-scale images in real-time, it is essential to design parallel algorithms that utilize distributed memory computers efficiently. The aim of this paper is to illustrate recent advances of domain decomposition methods for total variation minimization as parallel algorithms. Domain decomposition methods are suitable for parallel computation since they solve a large-scale problem by dividing it into smaller problems and treating them in parallel, and they already have been widely used in structural mechanics. Differently from problems arising in structural mechanics, energy functionals of total variation minimization problems are in general nonlinear, nonsmooth, and nonseparable. Hence, designing efficient domain decomposition methods for total variation minimization is a quite challenging issue. We describe various existing approaches on domain decomposition methods for total variation minimization in a unified view. We address how the direction of research on the subject has changed over the past few years, and suggest several interesting topics for further research.On the convergence of nonstationary column-oriented version of algebraic iterative methods.https://www.zbmath.org/1453.650702021-02-27T13:50:00+00:00"Karimpour, Mehdi"https://www.zbmath.org/authors/?q=ai:karimpour.mehdi"Nikazad, Touraj"https://www.zbmath.org/authors/?q=ai:nikazad.tourajSummary: Recently, \textit{T. Elfving} et al. [Numer. Algorithms 74, No. 3, 905--924 (2017; Zbl 1366.65116)] introduced a successful nonstationary block-column iterative method for solving linear system of equations based on flagging idea (called BCI-F). Their numerical tests show that the column-action method provides a basis for saving computational work using flagging technique in BCI algorithm. However, they did not present a general convergence analysis. In this paper, we give a convergence analysis of BCI-F. Furthermore, we consider a fully flexible version of block-column iterative method (FBCI), in which the relaxation parameters and weight matrices can be updated in each iteration and the column partitioning of coefficient matrix is allowed to update in each cycle. We also provide the convergence analysis of algorithm FBCI under mild conditions.A combined boundary element and finite element model of cell motion due to chemotaxis.https://www.zbmath.org/1453.653222021-02-27T13:50:00+00:00"Harris, Paul J."https://www.zbmath.org/authors/?q=ai:harris.paul-jSummary: Chemotaxis is the biological process whereby a cell moves in the direction in which the concentration of a chemical in the fluid medium surrounding the cell is increasing. In some cases of chemotaxis, cells secrete the chemical in order to create a concentration gradient that will attract other nearby cells to form clusters. When the cell secreting the chemical is stationary the linear diffusion equation can be used to model the concentration of the chemical as it spreads out into the surrounding fluid medium. However, if the cell is moving then its motion and the resulting motion of the surrounding fluid need to be taken into account in any model of how the chemical spreads out. In the case of a single, circular cell it is possible to express the fluid velocity which results from the motion of the cell in terms of a dipole located at the center of the cell. However if the cell is not circular and/or there is more than one cell in the fluid, a more sophisticated method of determining the fluid velocity is needed. This paper presents a mathematical model for simulating the concentrations of chemical secreted into the surrounding fluid medium from a moving cell. The boundary integral method is used to determine the velocity of the fluid due to the motion of the cell. The concentration of the chemical in the fluid is modelled by the convection-diffusion equation where the fluid velocity term is that given by the boundary integral equation. The resulting differential equation is then solved using the finite element method. The method is illustrated with a number of typical examples.
For the entire collection see [Zbl 1417.65006].Resource efficient finite element computing on multicore architectures.https://www.zbmath.org/1453.654102021-02-27T13:50:00+00:00"Kopysov, Sergeĭ Petrovich"https://www.zbmath.org/authors/?q=ai:kopysov.sergei-petrovich"Kadyrov, Il'yas Rinatovich"https://www.zbmath.org/authors/?q=ai:kadyrov.ilyas-rinatovich"Novikov, Aleksandr Konstantinovich"https://www.zbmath.org/authors/?q=ai:novikov.aleksandr-konstantinovichSummary: In this paper, we consider the construction of efficient finite element algorithms on three-dimensional unstructured meshes that take into account the complex parallel synchronization processes, the memory distribution problems and data storage. A layer-by-layer partitioning of the meshes into subdomains without branching internal boundaries is proposed to simplify the access to independent data and parallel computing at different stages of the finite element problem solving on unstructured meshes in multiply connected domains. The predictive capacity of the time efficiency and resource intensity for the proposed algorithmic solutions is analyzed. The analysis of the resource efficiency of the algorithms is given for the element-by-element scheme for forming and solving the system of linear algebraic equations of the finite element method. It is shown that the low arithmetic intensity of the algorithms considered results in the fact that their performance is limited by the bandwidth of the memory subsystem rather than by the processors' performance. The graphic memory has a larger bandwidth than the random-access memory. This allows a significant increase in the performance of the algorithm on GPU.Efficient and error minimized coupling procedures for unstructured and moving meshes.https://www.zbmath.org/1453.652732021-02-27T13:50:00+00:00"Lundquist, Tomas"https://www.zbmath.org/authors/?q=ai:lundquist.tomas"Malan, Arnaud G."https://www.zbmath.org/authors/?q=ai:malan.arnaud-george"Nordström, Jan"https://www.zbmath.org/authors/?q=ai:nordstrom.janSummary: We present a methodology for automatic generation and optimization of interpolation operators for the coupling of general non-collocated and/or moving numerical interfaces. The discrete equations are solved in a method-of-lines fashion by assuming volume preserving mesh motions. Interface interpolation errors are minimized effectively in a global least-squares sense, while satisfying strict stability conditions. The proposed automatic interface procedure is both more versatile and more accurate compared to previous techniques. We apply the new method to interfaces between hybrid meshes undergoing relative rigid body motion, demonstrating the stability, conservation and superior accuracy.Numerical methods for one-dimensional hyperbolic conservation laws.https://www.zbmath.org/1453.650052021-02-27T13:50:00+00:00"Tang, Huazhong"https://www.zbmath.org/authors/?q=ai:tang.huazhongConvex non-convex image segmentation.https://www.zbmath.org/1453.651412021-02-27T13:50:00+00:00"Chan, Raymond"https://www.zbmath.org/authors/?q=ai:chan.raymond-hon-fu"Lanza, Alessandro"https://www.zbmath.org/authors/?q=ai:lanza.alessandro"Morigi, Serena"https://www.zbmath.org/authors/?q=ai:morigi.serena"Sgallari, Fiorella"https://www.zbmath.org/authors/?q=ai:sgallari.fiorellaThe paper considers the variational model for multiphase segmentation of images in which the minimized energy functional consists of a standard strictly convex quadratic fidelity term and a non-convex regularization term designed for penalizing simultaneously the non-smoothness of the inner segmented regions and the length of the boundaries. Sufficient conditions for convexity are derived for the model. An iterative minimization procedure based on the alternating direction method of multipliers is proposed. The convergence of the algorithm is established. The performance of the procedure is demonstrated by three numerical examples.
Reviewer: Hang Lau (Montréal)Eighth-order, phase-fitted, four-step methods for solving \(y^{\prime \prime} = f(x, y)\).https://www.zbmath.org/1453.651562021-02-27T13:50:00+00:00"Alolyan, Ibraheem"https://www.zbmath.org/authors/?q=ai:alolyan.ibraheem"Simos, Theodore E."https://www.zbmath.org/authors/?q=ai:simos.theodore-e"Tsitouras, Charalampos"https://www.zbmath.org/authors/?q=ai:tsitouras.charalamposSummary: A family of explicit, eighth-order, four-step methods for the numerical solution of \(y^{\prime\prime} = f(x, y)\) is considered. This family is derived through an interpolatory approach after using three stages (ie, function evaluations) per step. In the present work, we alter three of the coefficients of such a method in order to become phase fitted. We conclude with numerical tests over a set of problems justifying our effort of dealing with the new methods.Error estimates of a conservative finite difference Fourier pseudospectral method for the Klein-Gordon-Schrödinger equation.https://www.zbmath.org/1453.652222021-02-27T13:50:00+00:00"Ji, Bingquan"https://www.zbmath.org/authors/?q=ai:ji.bingquan"Zhang, Luming"https://www.zbmath.org/authors/?q=ai:zhang.lumingSummary: A semi-linearized time-stepping method based on the finite difference in time and Fourier pseudospectral discretization for spatial derivatives is constructed and analyzed for the Klein-Gordon-Schrödinger equation. The resulting numerical scheme is proved to conserve the total mass and energy in the discrete levels. The maximum norm error estimates reflecting the second-order accuracy in time and spectral accuracy in space are established by using the standard energy method coupled with the induction argument. An efficient numerical algorithm is reported to speed up the evaluation of resulting algebra equation by means of the fast discrete Fourier transform. Numerical experiments are presented to show the effectiveness of our method and to confirm our analysis.Thermodynamic properties of Heisenberg spin systems on a square lattice with the Dzyaloshinskii-Moriya interaction.https://www.zbmath.org/1453.820012021-02-27T13:50:00+00:00"Kapitan, V. Yu."https://www.zbmath.org/authors/?q=ai:kapitan.v-yu"Vasil'ev, E. V."https://www.zbmath.org/authors/?q=ai:vasilev.e-v"Shevchenko, Yu. A."https://www.zbmath.org/authors/?q=ai:shevchenko.yu-a"Perzhu, A. V."https://www.zbmath.org/authors/?q=ai:perzhu.a-v"Kapitan, D. Yu."https://www.zbmath.org/authors/?q=ai:kapitan.d-yu"Rybin, A. E."https://www.zbmath.org/authors/?q=ai:rybin.a-e"Soldatov, K. S."https://www.zbmath.org/authors/?q=ai:soldatov.k-s"Makarov, A. G."https://www.zbmath.org/authors/?q=ai:makarov.a-g"Volotovskiĭ, R. A."https://www.zbmath.org/authors/?q=ai:volotovskii.r-a"Nefedev, K. V."https://www.zbmath.org/authors/?q=ai:nefedev.k-vSummary: We present results of numerical simulation of thermodynamics for array of Classical Heisenberg spins placed on 2D square lattice. By using Metropolis and Wang-Landau methods we show the temperature behaviour of system with competing Heisenberg and Dzyaloshinskii-Moriya interaction (DMI) in contrast with classical Heisenberg system. We show the process of nucleating of skyrmion depending on the value of external magnetic field.An efficient algorithm for a class of stochastic forward and inverse Maxwell models in \(\mathbb{R}^3\).https://www.zbmath.org/1453.650172021-02-27T13:50:00+00:00"Ganesh, M."https://www.zbmath.org/authors/?q=ai:ganesh.mahadevan"Hawkins, S. C."https://www.zbmath.org/authors/?q=ai:hawkins.stuart-collin"Volkov, D."https://www.zbmath.org/authors/?q=ai:volkov.darkoSummary: We describe an efficient algorithm for reconstruction of the electromagnetic parameters of an unbounded dielectric medium from noisy cross section data induced by a point source in \(\mathbb{R}^3\). The efficiency of our Bayesian inverse algorithm for the parameters is based on developing an offline high order forward stochastic model and also an associated deterministic dielectric media Maxwell solver. Underlying the inverse/offline approach is our high order fully discrete Galerkin algorithm for solving an equivalent surface integral equation reformulation that is stable for all frequencies. The efficient algorithm includes approximating the likelihood distribution in the Bayesian model by a decomposed fast generalized polynomial chaos (gPC) model as a surrogate for the forward model. Offline construction of the gPC model facilitates fast online evaluation of the posterior distribution of the dielectric medium parameters. Parallel computational experiments demonstrate the efficiency of our deterministic, forward stochastic, and inverse dielectric computer models.Parallel cluster multiple labeling technique.https://www.zbmath.org/1453.820352021-02-27T13:50:00+00:00"Lapshina, S. Yu."https://www.zbmath.org/authors/?q=ai:lapshina.s-yuSummary: A parallel cluster multiple labeling technique, which allows simulation experiments on multiprocessor computing systems is considered in this paper. This technique belongs to the class of algorithms for substructures of a percolation cluster by definition of the percolation threshold and sizes of percolation clusters.Models and numerical methods for electrolyte flows.https://www.zbmath.org/1453.652502021-02-27T13:50:00+00:00"Fuhrmann, Jürgen"https://www.zbmath.org/authors/?q=ai:fuhrmann.jurgen"Guhlke, Clemens"https://www.zbmath.org/authors/?q=ai:guhlke.clemens"Linke, Alexander"https://www.zbmath.org/authors/?q=ai:linke.alexander"Merdon, Christian"https://www.zbmath.org/authors/?q=ai:merdon.christian"Müller, Rüdiger"https://www.zbmath.org/authors/?q=ai:muller.rudigerSummary: Liquid electrolytes are fluidic mixtures containing electrically charged ions. Electrochemical energy conversion systems like fuel cells and batteries contain liquid electrolytes. In biological tissues, nanoscale pores in the cell membranes separate different types of ions inside the cell from those in the intercellular space. Nanopores between electrolyte reservoirs can be used for analytical applications in medicine. Water purification technologies like electrodialysis rely on the electrolytic flow properties. This short and by far not exhaustive list of occurrences of electrolytic flow processes shows the importance of correct modeling of electrolyte flows. Due to the complex physical interactions present in this type of flows, in many case numerical simulation techniques are required to facilitate a deeper understanding of the flow behavior.
For the entire collection see [Zbl 1433.35004].Solving the inverse problem for an ordinary differential equation using conjugation.https://www.zbmath.org/1453.340242021-02-27T13:50:00+00:00"Alfaro Vigo, Daniel G."https://www.zbmath.org/authors/?q=ai:alfaro-vigo.daniel-gregorio|alfaro-vigo.daniel-g"Álvarez, Amaury C."https://www.zbmath.org/authors/?q=ai:alvarez.amaury-c"Chapiro, Grigori"https://www.zbmath.org/authors/?q=ai:chapiro.grigori"García, Galina C."https://www.zbmath.org/authors/?q=ai:garcia.galina-c"Moreira, Carlos G."https://www.zbmath.org/authors/?q=ai:moreira.carlos-gustavo-t-de-aThe authors consider the inverse problem of recovering an ordinary differential equation \(x'(t) = v(x)\) from a set of data points \(P = \{(t_i, x_i), i = 1,\dots, N\}\), such that \(x_i\approx x(t_i)\) as closely as possible. The key to the proposed method is to find approximations of the recursive or discrete propagation function \(D(x)\) from the given data set. The field \(v(x)\) is determined using the conjugate map defined by Schröder's equation and the solution of a related Julia's equation. Numerical algorithms are presented at the end.
Reviewer: Amin Boumenir (Carrollton)An improvement of third order WENO scheme for convergence rate at critical points with new non-linear weights.https://www.zbmath.org/1453.652242021-02-27T13:50:00+00:00"Kumar, Anurag"https://www.zbmath.org/authors/?q=ai:kumar.anurag"Kaur, Bhavneet"https://www.zbmath.org/authors/?q=ai:kaur.bhavneetSummary: In this paper, we construct and implement a new improvement of third order weighted essentially non-oscillatory (WENO) scheme in the finite difference framework for hyperbolic conservation laws. In our approach, a modification in the global smoothness measurement is reported by applying all three points on global stencil \((i-1,i,i+1)\) which is used for convergence of non-linear weights towards the optimal weights at critical points and achieves the desired order of accuracy for third order WENO scheme. We use the third order accurate total variation diminishing (TVD) Runge-Kutta time stepping method. The major advantage of the proposed scheme is its better numerical accuracy in smooth regions. The computational performance of the proposed WENO scheme with this global smoothness measurement is verified in several benchmark one- and two-dimensional test cases for scalar and vector hyperbolic equations. Extensive computational results confirm that the new proposed scheme achieves better performance as compared with WENO-JS3, WENO-Z3 and WENO-F3 schemes.Connectivity of the Julia set for the Chebyshev-Halley family on degree \(n\) polynomials.https://www.zbmath.org/1453.370392021-02-27T13:50:00+00:00"Campos, B."https://www.zbmath.org/authors/?q=ai:campos.beatriz"Canela, J."https://www.zbmath.org/authors/?q=ai:canela.jordi"Vindel, P."https://www.zbmath.org/authors/?q=ai:vindel.puraSummary: We study the Chebyshev-Halley family of root finding algorithms from the point of view of holomorphic dynamics. In this paper we provide a criterion which guarantees the simple connectivity of the basins of attraction of the roots. We use the criterion for the Chebyshev-Halley methods applied to the degree \(n\) polynomials \(z^n + c\), obtaining a characterization of the parameters for which all Fatou components are simply connected and, therefore, the Julia set is connected. We also study how increasing \(n\) affects the dynamics.Metric-like spaces to prove existence of solution for nonlinear quadratic integral equation and numerical method to solve it.https://www.zbmath.org/1453.450022021-02-27T13:50:00+00:00"Hazarika, Bipan"https://www.zbmath.org/authors/?q=ai:hazarika.bipan"Karapinar, Erdal"https://www.zbmath.org/authors/?q=ai:karapinar.erdal"Arab, Reza"https://www.zbmath.org/authors/?q=ai:arab.reza"Rabbani, Mohsen"https://www.zbmath.org/authors/?q=ai:rabbani.mohsenSummary: The purpose of this paper is to obtain some common fixed point results for two mappings satisfying various contractive conditions in metric-like spaces. These results extend some previous results in the literature, since the condition under which the operator admits common fixed points is more general than the others in literature. Therefore, several well known results are generalized. As an application we use these results to existence of solution for nonlinear quadratic integral equation. To credibility, we apply modified homotopy and Adomian decomposition method to find solution of the above problem with high accuracy.Efficient preconditioner updates for semilinear space-time fractional reaction-diffusion equations.https://www.zbmath.org/1453.652842021-02-27T13:50:00+00:00"Bertaccini, Daniele"https://www.zbmath.org/authors/?q=ai:bertaccini.daniele"Durastante, Fabio"https://www.zbmath.org/authors/?q=ai:durastante.fabioSummary: The numerical solution of fractional partial differential equations poses significant computational challenges in regard to efficiency as a result of the nonlocality of the fractional differential operators. In this work we consider the numerical solution of nonlinear space-time fractional reaction-diffusion equations integrated in time by fractional linear multistep formulas. The Newton step needed to advance in (fractional) time requires the solution of sequences of large and dense linear systems because of the fractional operators in space. A preconditioning updating strategy devised recently is adapted and the spectrum of the underlying operators is briefly analyzed. Because of the quasilinearity of the problem, each Jacobian matrix of the Newton equations can be written as the sum of a multilevel Toeplitz plus a diagonal matrix and produced exactly in the code. Numerical tests with a population dynamics problem show that the proposed approach is fast and reliable with respect to standard direct, unpreconditioned, multilevel circulant/Toeplitz and ILU preconditioned iterative solvers.
For the entire collection see [Zbl 1416.65008].Numerical simulation of inextensible elastic ribbons.https://www.zbmath.org/1453.740032021-02-27T13:50:00+00:00"Bartels, Sören"https://www.zbmath.org/authors/?q=ai:bartels.sorenElastic ribbons are thin elastic bands of small width, which have a range of applications, particularly in nanotechnology. Dimensionally reduced models are necessary to effectively simulate the highly nonlinear behavior of thin elastic objects. In this paper, a one-dimensional model of ribbons is discretized with the finite elements by minimizing the energy functional. Because the energy functional is continuously differentiable but not twice continuously differentiable, regularization is required. The iterative minimization should provide reliably decrease of the energy and avoid the occurrence of irregular stationary configurations. A convergent finite element discrete approximation of the regularized variational problem is devised by respecting these issues and its convergence to the continuous model is investigated. Numerical experiments with the Möbius ribbon and twisted helix demonstrate the computational efficiency and accuracy of the proposed approach.
Reviewer: Bülent Karasözen (Ankara)An improved firefly algorithm for numerical optimisation.https://www.zbmath.org/1453.651442021-02-27T13:50:00+00:00"Xiang, Xiangqin"https://www.zbmath.org/authors/?q=ai:xiang.xiangqinSummary: Firefly algorithm (FA) is a recently proposed meta-heuristic optimisation technique, which has shown good performance on many optimisation problems. In the original FA, each firefly is attracted by any other brighter firefly (better fitness value). By the attraction, fireflies maybe moved to better positions. However, the attraction does not guarantee whether a firefly is moved to a better position. Sometimes, the attraction may move a firefly to a worse position. Therefore, the search of firefly is oscillated during the evolution. In this paper, we present an improved firefly algorithm (IFA), which employs a greedy selection method to guarantee that a firefly is not moved to worse positions. To verify the performance of IFA, a set of well-known benchmark functions are used in the experiments. Experimental results show that the IFA achieves better results than the original FA.New proper orthogonal decomposition approximation theory for PDE solution data.https://www.zbmath.org/1453.651142021-02-27T13:50:00+00:00"Locke, Sarah"https://www.zbmath.org/authors/?q=ai:locke.sarah"Singler, John"https://www.zbmath.org/authors/?q=ai:singler.john-rFar field boundary conditions for incompressible flows computation.https://www.zbmath.org/1453.653032021-02-27T13:50:00+00:00"Bruneau, Charles-Henri"https://www.zbmath.org/authors/?q=ai:bruneau.charles-henri"Tancogne, Sandra"https://www.zbmath.org/authors/?q=ai:tancogne.sandraSummary: Many far field boundary conditions are proposed in the literature to solve Navier-Stokes equations. It is necessary to distinguish the streamwise or outlet boundary conditions and the spanwise boundary conditions. In the first case the flow crosses the artificial frontier and it is required to avoid reflections that can change significantly the flow. In the second case the Navier-slip boundary condition is often used but if the frontier is not far enough the boundary is both inlet and outlet. Thus the Navier-slip boundary condition is not well suited as it imposes no flux through the frontier. The aim of this work is to compare some well-known boundary conditions, to quantify to which extend the artificial frontier can be close to the bodies in two- and three-dimensions and to take into account the flow rate through the spanwise directions.Two-derivative error inhibiting schemes and enhanced error inhibiting schemes.https://www.zbmath.org/1453.651642021-02-27T13:50:00+00:00"Ditkowski, Adi"https://www.zbmath.org/authors/?q=ai:ditkowski.adi"Gottlieb, Sigal"https://www.zbmath.org/authors/?q=ai:gottlieb.sigal"Grant, Zachary J."https://www.zbmath.org/authors/?q=ai:grant.zachary-jOperational matrices of Chebyshev polynomials for solving singular Volterra integral equations.https://www.zbmath.org/1453.654632021-02-27T13:50:00+00:00"Sahlan, Monireh Nosrati"https://www.zbmath.org/authors/?q=ai:sahlan.monireh-nosrati"Feyzollahzadeh, Hadi"https://www.zbmath.org/authors/?q=ai:feyzollahzadeh.hadiSummary: An effective technique based on fractional calculus in the sense of Riemann-Liouville has been developed for solving weakly singular Volterra integral equations of the first and second kinds. For this purpose, orthogonal Chebyshev polynomials are applied. Properties and some operational matrices of these polynomials are first presented and then the unknown functions of the integral equations are represented by these polynomials in the matrix form. These matrices are then used to reduce the singular integral equations to some linear algebraic system. For solving the obtained system, Galerkin method is utilized via Chebyshev polynomials as weighting functions. The method is computationally attractive, and the validity and accuracy of the presented method are demonstrated through illustrative examples. As shown in the numerical results, operational matrices, even for first kind integral equations, have relatively low condition numbers, and thus, the corresponding matrices are well posed. In addition, it is noteworthy that when the solution of equation is in power series form, the method evaluates the exact solution.Wavelet methods for solving three-dimensional partial differential equations.https://www.zbmath.org/1453.654232021-02-27T13:50:00+00:00"Singh, Inderdeep"https://www.zbmath.org/authors/?q=ai:singh.inderdeep"Kumar, Sheo"https://www.zbmath.org/authors/?q=ai:kumar.sheoSummary: We present, a collocation method based on Haar wavelet and Kronecker tensor product for solving three-dimensional partial differential equations. The method is based on approximating a sixth-order mixed derivative by a series of Haar wavelet basis functions. The present method is suitable for numerical solution of all kinds of three-dimensional Poisson and Helmholtz equations. Numerical examples are solving to establish the efficiency and accuracy of the present method. Numerical results obtained are better as compared to numerical results obtained in past.Numerical solution of nonlinear two-dimensional Volterra integral equation of the second kind in the reproducing kernel space.https://www.zbmath.org/1453.654512021-02-27T13:50:00+00:00"Fazli, A."https://www.zbmath.org/authors/?q=ai:fazli.a"Allahviranloo, T."https://www.zbmath.org/authors/?q=ai:allahviranloo.tofigh"Javadi, Sh."https://www.zbmath.org/authors/?q=ai:javadi.shahnamSummary: In this article, an effective method is given to solve nonlinear two-dimensional Volterra integral equations of the second kind. First, we find the solution of integral equation in terms of reproducing kernel functions in series, then by truncating the series an approximate solution obtained. In addition, the calculation of Fourier coefficients solution of the integral equation in terms of reproducing kernel functions is notable. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the method.Close encounters of the binary kind: signal reconstruction guarantees for compressive Hadamard sampling with Haar wavelet basis.https://www.zbmath.org/1453.940312021-02-27T13:50:00+00:00"Moshtaghpour, Amirafshar"https://www.zbmath.org/authors/?q=ai:moshtaghpour.amirafshar"Bioucas-Dias, José M."https://www.zbmath.org/authors/?q=ai:bioucas-dias.jose-m"Jacques, Laurent"https://www.zbmath.org/authors/?q=ai:jacques.laurentEditorial remark: No review copy delivered.An iterative algorithm to solve the generalized Sylvester tensor equations.https://www.zbmath.org/1453.650842021-02-27T13:50:00+00:00"Huang, Baohua"https://www.zbmath.org/authors/?q=ai:huang.baohua"Ma, Changfeng"https://www.zbmath.org/authors/?q=ai:ma.changfengSummary: This paper is concerned with the conjugate gradient least squares algorithm to solve a class of tensor equations via the Einstein product. The proposed algorithm uses tensor computations with no matricizations involved. We prove that the solution (or the least squares solution) of the tensor equation can be obtained within a finite number of iterative steps in the absence of round-off errors. By selecting the appropriate initial tensor, the least Frobenius norm solution (or the least Frobenius norm least squares solution) of the tensor equation can be obtained. Numerical examples are provided to illustrate the efficiency of the proposed algorithm and testify the conclusions suggested in this paper.Extension of some theorems to find solution of nonlinear integral equation and homotopy perturbation method to solve it.https://www.zbmath.org/1453.654622021-02-27T13:50:00+00:00"Rabbani, Mohsen"https://www.zbmath.org/authors/?q=ai:rabbani.mohsen"Arab, Reza"https://www.zbmath.org/authors/?q=ai:arab.rezaSummary: In this paper, the concept of contraction via the measure of non-compactness on a Banach space is investigated by generalizing some results which have been previously discussed in literatures. Furthermore, to validity of the theorems and homotopy perturbation method (HPM), as a technical solution, they are applied on some nonlinear singular integral equations.A coupled implicit-explicit time integration method for compressible unsteady flows.https://www.zbmath.org/1453.761042021-02-27T13:50:00+00:00"Muscat, Laurent"https://www.zbmath.org/authors/?q=ai:muscat.laurent"Puigt, Guillaume"https://www.zbmath.org/authors/?q=ai:puigt.guillaume"Montagnac, Marc"https://www.zbmath.org/authors/?q=ai:montagnac.marc"Brenner, Pierre"https://www.zbmath.org/authors/?q=ai:brenner.pierreSummary: This paper addresses how two time integration schemes, the Heun's scheme for explicit time integration and the second-order Crank-Nicolson scheme for implicit time integration, can be coupled spatially. This coupling is the prerequisite to perform a coupled Large Eddy Simulation/Reynolds Averaged Navier-Stokes computation in an industrial context, using the implicit time procedure for the boundary layer (RANS) and the explicit time integration procedure in the LES region. The coupling procedure is designed in order to switch from explicit to implicit time integrations as fast as possible, while maintaining stability. After introducing the different schemes, the paper presents the initial coupling procedure adapted from a published reference and shows that it can amplify some numerical waves. An alternative procedure, studied in a coupled time/space framework, is shown to be stable and with spectral properties in agreement with the requirements of industrial applications. The coupling technique is validated with standard test cases, ranging from one-dimensional to three-dimensional flows.A finite difference discretization method for heat and mass transfer with Robin boundary conditions on irregular domains.https://www.zbmath.org/1453.652112021-02-27T13:50:00+00:00"Chai, Min"https://www.zbmath.org/authors/?q=ai:chai.min"Luo, Kun"https://www.zbmath.org/authors/?q=ai:luo.kun"Shao, Changxiao"https://www.zbmath.org/authors/?q=ai:shao.changxiao"Wang, Haiou"https://www.zbmath.org/authors/?q=ai:wang.haiou"Fan, Jianren"https://www.zbmath.org/authors/?q=ai:fan.jianrenSummary: This paper proposes a finite difference discretization method for simulations of heat and mass transfer with Robin boundary conditions on irregular domains. The level set method is utilized to implicitly capture the irregular evolving interface, and the ghost fluid method to address variable discontinuities on the interface. Special care has been devoted to providing ghost values that are restricted by the Robin boundary conditions. Specifically, it is done in two steps: 1) calculate the normal derivative in cells adjacent to the interface by reconstructing a linear polynomial system; 2) successively extrapolate the normal derivative and the ghost value in the normal direction using a linear partial differential equation approach. This method produces second-order accurate solutions for both the Poisson and heat equations with Robin boundary conditions, and first-order accurate solutions for the Stefan problems. The solution gradients are of first-order accuracy, as expected. It is easy to implement in three-dimensional configurations, and can be straightforwardly generalized into higher-order variants. The method thus represents a promising tool for practical heat and mass transfer problems involving Robin boundary conditions.An efficient step method for a system of differential equations with delay.https://www.zbmath.org/1453.470232021-02-27T13:50:00+00:00"Otrocol, Diana"https://www.zbmath.org/authors/?q=ai:otrocol.diana"Serban, Marcel-Adrian"https://www.zbmath.org/authors/?q=ai:serban.marcel-adrianSummary: Using the step method, we study a system of delay differential equations and we prove the existence and uniqueness of the solution and the convergence of the successive approximation sequence using Perov's contraction principle and the step method. Also, we propose a new algorithm of successive approximation sequence generated by the step method and, as an example, we consider some second order delay differential equations with initial conditions.A connection between coupled and penalty projection timestepping schemes with FE spatial discretization for the Navier-Stokes equations.https://www.zbmath.org/1453.653312021-02-27T13:50:00+00:00"Linke, Alexander"https://www.zbmath.org/authors/?q=ai:linke.alexander"Neilan, Michael"https://www.zbmath.org/authors/?q=ai:neilan.michael"Rebholz, Leo G."https://www.zbmath.org/authors/?q=ai:rebholz.leo-g"Wilson, Nicholas E."https://www.zbmath.org/authors/?q=ai:wilson.nicholas-eSummary: We prove that for several inf-sup stable mixed finite elements, the solution of the Chorin/Temam projection methods for Navier-Stokes equations equipped with grad-div stabilization with parameter \(\gamma\) converge to the associated coupled method solution with rate \(\gamma^{-1}\) as \(\gamma\rightarrow\infty\). We prove this result for both backward Euler schemes and BDF2 schemes. Furthermore, we simplify classical numerical analysis of projection methods, allowing us to remove some unnecessary assumptions, such as convexity of the domain. Several numerical experiments are given which verify the convergence rate, and show that projection methods with large grad-div stabilization parameters can dramatically improve accuracy.A fully Eulerian finite volume method for the simulation of fluid-structure interactions on AMR enabled quadtree grids.https://www.zbmath.org/1453.652442021-02-27T13:50:00+00:00"Bergmann, Michel"https://www.zbmath.org/authors/?q=ai:bergmann.michel"Fondanèche, Antoine"https://www.zbmath.org/authors/?q=ai:fondaneche.antoine"Iollo, Angelo"https://www.zbmath.org/authors/?q=ai:iollo.angeloThe authors propose a finite-volume scheme for solving a single-continuum model of fluid-structure interactions. The numerical Rusanov (Local Lax-Friedrichs) flux is introduced to compute the convective/transport flux. In this formulation, the stabilization is simply performed according to the normal face-center velocity without considering the velocity of the waves which propagate inside the material. The finite volume scheme is stable only for moderately stiff material or for high viscosities. An example of application is presented and detailed.
For the entire collection see [Zbl 1445.65003].
Reviewer: Abdallah Bradji (Annaba)Semi-implicit two-speed well-balanced relaxation scheme for Ripa model.https://www.zbmath.org/1453.652492021-02-27T13:50:00+00:00"Franck, Emmanuel"https://www.zbmath.org/authors/?q=ai:franck.emmanuel"Navoret, Laurent"https://www.zbmath.org/authors/?q=ai:navoret.laurentSummary: In this paper, we propose a semi-implicit well-balanced scheme for the Ripa model based on a two-speed relaxation. The method both preserves equilibria and has an implicit step that reduces to the inversion of a constant Laplacian. Numerical simulations show that the scheme well capture low-Froude flows.
For the entire collection see [Zbl 1445.65003].Estimating failure probabilities.https://www.zbmath.org/1453.653752021-02-27T13:50:00+00:00"ter Maten, E. Jan W."https://www.zbmath.org/authors/?q=ai:ter-maten.e-jan-w"Beelen, Theo G. J."https://www.zbmath.org/authors/?q=ai:beelen.theo-g-j"Di Bucchianico, Alessandro"https://www.zbmath.org/authors/?q=ai:di-bucchianico.alessandro"Pulch, Roland"https://www.zbmath.org/authors/?q=ai:pulch.roland"Römer, Ulrich"https://www.zbmath.org/authors/?q=ai:romer.ulrich"De Gersem, Herbert"https://www.zbmath.org/authors/?q=ai:de-gersem.herbert"Janssen, Rick"https://www.zbmath.org/authors/?q=ai:janssen.rick"Dohmen, Jos J."https://www.zbmath.org/authors/?q=ai:dohmen.jos-j"Tasić, Bratislav"https://www.zbmath.org/authors/?q=ai:tasic.bratislav"Gillon, Renaud"https://www.zbmath.org/authors/?q=ai:gillon.renaud"Wieers, Aarnout"https://www.zbmath.org/authors/?q=ai:wieers.aarnout"Deleu, Frederik"https://www.zbmath.org/authors/?q=ai:deleu.frederikSummary: System failure describes an undesired configuration of an engineering device, possibly leading to the destruction of material or a significant loss of performance and a consequent loss of yield. For systems subject to uncertainties, failure probabilities express the probability of this undesired configuration to take place. The accurate computation of failure probabilities, however, can be very difficult in practice. It may also become very costly, because of the many Monte Carlo samples that have to be taken, which may involve time consuming evaluations. In this chapter we present an overview of techniques to realistically estimate the amount of Monte Carlo runs that are needed to guarantee sharp bounds for relative errors of failure probabilities. They are presented for Monte Carlo sampling and for Importance Sampling. These error estimates apply to both non-parametric and parametric sampling. In the case of parametric sampling we propose a hybrid algorithm that combines simulations of full models and approximating response surface models. We illustrate this hybrid algorithm with a computation of bond wire fusing probabilities.
For the entire collection see [Zbl 1433.78001].Spectral collocation solutions to problems on unbounded domains.https://www.zbmath.org/1453.650042021-02-27T13:50:00+00:00"Gheorghiu, Călin-Ioan"https://www.zbmath.org/authors/?q=ai:gheorghiu.calin-ioanA usual procedure in discretizing problems on unbounded domains is to cut off an exterior part of the domain, to impose suitable boundary conditions on the newly obtained boundary and then to apply methods designed for problems on bounded domains. The author instead discretizes the problems directly as given by spectral approximation with trial functions defined on the entire domain combined with collocation in a finite number of points. As trial function he considers Hermite functions, Laguerre functions (with both the roots of the Laguerre polynomials and the Gauss-Radau nodes as collocation points) and sinc functions and also a mapping technique. The trial functions depend on a scaling parameter which is also adapted for obtaining good results.
The author concentrates on presenting a large number of numerical results, an analysis of the methods is not in the focus. The book contains 105 figures and 23 tables (a list of problems considered can be found under the above keywords). The calculations are performed using MATLAB (for some representative problems MATLAB scripts are given in the fifth chapter of the book). The bibliography comprises 195 items.
The book is for sale at 19.99 Euro. But I think that this affordable price is not the explanation for the not so successful layout of the figures. They seem to be placed at random in the center, at the left margin, colored or not, with weakly or firmly printed labels and lines.
Reviewer: Rolf Dieter Grigorieff (Berlin)Errors-in-variables and the wavelet multiresolution approximation approach: a Monte Carlo study.https://www.zbmath.org/1453.624772021-02-27T13:50:00+00:00"Gallegati, Marco"https://www.zbmath.org/authors/?q=ai:gallegati.marco"Ramsey, James B."https://www.zbmath.org/authors/?q=ai:ramsey.james-bSummary: In this chapter we perform a Monte Carlo simulation study of the errors-in-variables model examined in [\textit{J. B. Ramsey}, \textit{M. Gallegati}, \textit{M. Gallegati} and \textit{W. Semmler}, ``Instrumental variables and wavelet decomposition'', Economic Modeling 27, No. 6, 1498--1513 (2010; \url{doi:10.1016/j.econmod.2010.07.011})] by using a wavelet multiresolution approximation approach. Differently from previous studies applying wavelets to errors-in-variables problem, we use a sequence of multiresolution approximations of the variable measured with error ranging from finer to coarser scales. Our results indicate that multiscale approximations to the variable observed with error based on the coarser scales provide an unbiased asymptotically efficient estimator that also possess good finite sample properties.
For the entire collection see [Zbl 1298.91020].On the structure of exchangeable extreme-value copulas.https://www.zbmath.org/1453.600422021-02-27T13:50:00+00:00"Mai, Jan-Frederik"https://www.zbmath.org/authors/?q=ai:mai.jan-frederik"Scherer, Matthias"https://www.zbmath.org/authors/?q=ai:scherer.matthiasSummary: We show that the set of \(d\)-variate symmetric stable tail dependence functions is a simplex and we determine its extremal boundary. The subset of elements which arises as \(d\)-margins of the set of \(( d + k )\)-variate symmetric stable tail dependence functions is shown to be proper for arbitrary \(k \geq 1\). Finally, we derive an intuitive and useful necessary condition for a bivariate extreme-value copula to arise as bi-margin of an exchangeable extreme-value copula of arbitrarily large dimension, and thus to be conditionally iid.A modified fifth order finite difference Hermite WENO scheme for hyperbolic conservation laws.https://www.zbmath.org/1453.652412021-02-27T13:50:00+00:00"Zhao, Zhuang"https://www.zbmath.org/authors/?q=ai:zhao.zhuang"Zhang, Yong-Tao"https://www.zbmath.org/authors/?q=ai:zhang.yongtao"Qiu, Jianxian"https://www.zbmath.org/authors/?q=ai:qiu.jianxianThe authors present a new modified fifth-order finite difference Hermite weighted essentially non-oscillatory (HWENO) scheme for solving one-dimensional and two-dimensional hyperbolic conservation laws. One of the main features of this scheme is that the HWENO scheme is simpler and more robust. In particular, it is not required to add a positivity-preserving flux limiter, and larger CFL number can be applied than the usual to ensure the stability of schemes. The new HWENO scheme has fifth-order accuracy while based on the same reconstruction stencil the original finite difference HWENO scheme only has fourth-order accuracy when solving two-dimensional problems. Furthermore, the new scheme preserves the nice property of compactness, i.e., only immediate neighbor information is needed in the reconstruction. Several numerical tests for both one-dimensional and two-dimensional problems are presented to illustrate the numerical accuracy, high resolution and robustness of the proposed modified HWENO scheme.
Reviewer: Abdallah Bradji (Annaba)On the partial synchronization of iterative methods.https://www.zbmath.org/1453.651062021-02-27T13:50:00+00:00"Dmitriev, A. V."https://www.zbmath.org/authors/?q=ai:dmitriev.anton-vladimirovich"Ermakov, S. M."https://www.zbmath.org/authors/?q=ai:ermakov.sergei-mikhailovichSummary: The rapidly growing field of parallel computing systems promotes the study of parallel algorithms, with the Monte Carlo method and asynchronous iterations being among the most valuable types. These algorithms have a number of advantages. There is no need for a global time in a parallel system (no need for synchronization), and all computational resources are efficiently loaded (the minimum processor idle time). The method of partial synchronization of iterations for systems of equations was proposed by the authors earlier. In this article, this method is generalized to include the case of nonlinear equations of the form \(x = F(x)\), where \(x\) is an unknown column vector of length \(n\), and \(F\) is an operator from \(\mathbb{R}^n\) into \(\mathbb{R}^n\). We consider operators that do not satisfy conditions that are sufficient for the convergence of asynchronous iterations, with simple iterations still converging. In this case, one can specify such an incidence of the operator and such properties of the parallel system that asynchronous iterations fail to converge. Partial synchronization is one of the effective ways to solve this problem. An algorithm is proposed that guarantees the convergence of asynchronous iterations and the Monte Carlo method for the above class of operators. The rate of convergence of the algorithm is estimated. The results can prove useful for solving high-dimensional problems on multiprocessor computational systems.A hybrid Hermite WENO scheme for hyperbolic conservation laws.https://www.zbmath.org/1453.652642021-02-27T13:50:00+00:00"Zhao, Zhuang"https://www.zbmath.org/authors/?q=ai:zhao.zhuang"Chen, Yibing"https://www.zbmath.org/authors/?q=ai:chen.yibing"Qiu, Jianxian"https://www.zbmath.org/authors/?q=ai:qiu.jianxianSummary: In this paper, we propose a hybrid finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for solving one and two dimensional hyperbolic conservation laws, which would be the fifth order accuracy in the one dimensional case, while is the fourth order accuracy for two dimensional problems. The zeroth-order and the first-order moments are used in the spatial reconstruction, with total variation diminishing Runge-Kutta time discretization. Unlike the original HWENO schemes [\textit{J. Qiu} and \textit{C.-W. Shu}, J. Comput. Phys. 193, No. 1, 115--135 (2004; Zbl 1039.65068); Comput. Fluids 34, No. 6, 642--663 (2005; Zbl 1134.65358)] using different stencils for spatial discretization, we borrow the thought of limiter for discontinuous Galerkin (DG) method to control the spurious oscillations, after this procedure, the scheme would avoid the oscillations by using HWENO reconstruction nearby discontinuities, and using linear approximation straightforwardly in the smooth regions is to increase the efficiency of the scheme. Moreover, the scheme still keeps the compactness as only immediate neighbor information is needed in the reconstruction. A collection of benchmark numerical tests for one and two dimensional cases are performed to demonstrate the numerical accuracy, high resolution and robustness of the proposed scheme.Least-squares spectral element preconditioners for fourth order elliptic problems.https://www.zbmath.org/1453.654222021-02-27T13:50:00+00:00"Husain, Akhlaq"https://www.zbmath.org/authors/?q=ai:husain.akhlaq"Khan, Arbaz"https://www.zbmath.org/authors/?q=ai:khan.arbazSummary: In this paper, we propose preconditioners for the system of linear equations that arise from a discretization of fourth order elliptic problems in two and three dimensions \((d=2,3)\) using spectral element methods. These preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert. For second order elliptic problems this technique has proven to be successful and performs better than other preconditioners in the framework of least-squares methods. We show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them. Numerical results for the condition number reflects the effectiveness of the preconditioners.Modified approach to solve nonlinear equation arising in infiltration phenomenon.https://www.zbmath.org/1453.761892021-02-27T13:50:00+00:00"Shah, Kunjan"https://www.zbmath.org/authors/?q=ai:shah.kunjan"Singh, Twinkle"https://www.zbmath.org/authors/?q=ai:singh.twinkle-rSummary: In the present analysis, the modified homotopy analysis method has been employed to find an approximate analytical solution of Richards' equation. This method is the slight modification of standard homotopy analysis method. Some standard cases of Richards' equation have been discussed as an example to illustrate the high accuracy and reliability of modified homotopy analysis method. The result obtained from the proposed method is very close to the exact solution of the problem. It is concluded that modified homotopy analysis method is the better alternative to some standard existing methods to solve some realistic problems arising in science and technology.Modeling and sensitivity analysis methodology for hybrid dynamical system.https://www.zbmath.org/1453.930702021-02-27T13:50:00+00:00"Corner, Sebastien"https://www.zbmath.org/authors/?q=ai:corner.sebastien"Sandu, Corina"https://www.zbmath.org/authors/?q=ai:sandu.corina"Sandu, Adrian"https://www.zbmath.org/authors/?q=ai:sandu.adrianThe paper is devoted to the direct sensitivity analysis for hybrid dynamical systems described by the second order ordinary differential equations or by differential algebraic equations with assotiated cost functions. For hybrid system piecewise-smooth trajectories with continuous position state variable and piecewise-continuous velocity state variable are allowed. The authors review the direct sensitivity analysis for unconsrained systems determined by ordinary differential equations and for smooth constrained multibody systems modelled by differential algebraic equations and generalize the direct sensitivity approach to hybrid systems. Graphic representation of the jumps in the sensitivity of the position and in the sensitivity of the velocity is proposed. The authors illustrate the developed approach for hybrid constrained multibody systems on the five-bar mechanism with two degrees of freedom.
Reviewer: Svetlana A. Kravchenko (Minsk)Incomplete iterative solution of subdiffusion.https://www.zbmath.org/1453.653262021-02-27T13:50:00+00:00"Jin, Bangti"https://www.zbmath.org/authors/?q=ai:jin.bangti"Zhou, Zhi"https://www.zbmath.org/authors/?q=ai:zhou.zhiSummary: In this work, we develop an efficient incomplete iterative scheme for the numerical solution of the subdiffusion model involving a Caputo derivative of order \(\alpha \in (0,1)\) in time. It is based on piecewise linear Galerkin finite element method in space and backward Euler convolution quadrature in time and solves one linear algebraic system inexactly by an iterative algorithm at each time step. We present theoretical results for both smooth and nonsmooth solutions, using novel weighted estimates of the time-stepping scheme. The analysis indicates that with the number of iterations at each time level chosen properly, the error estimates are nearly identical with that for the exact linear solver, and the theoretical findings provide guidelines on the choice. Illustrative numerical results are presented to complement the theoretical analysis.Extra-optimal methods for solving ill-posed problems: survey of theory and examples.https://www.zbmath.org/1453.651262021-02-27T13:50:00+00:00"Leonov, A. S."https://www.zbmath.org/authors/?q=ai:leonov.alexander-sSummary: A new direction in methods for solving ill-posed problems, namely, the theory of regularizing algorithms with approximate solutions of extra-optimal quality is surveyed. A distinctive feature of these methods is that they are optimal not only in the order of accuracy of resulting approximate solutions, but also with respect to a user-specified quality functional. Such functionals can be specified, for example, as an a posteriori estimate of the quality (accuracy) of approximate solutions, a posteriori estimates of various linear functionals of these solutions, and estimates of their mathematical entropy and multidimensional variations of chosen types. The relationship between regularizing algorithms that are extra-optimal and optimal in the order of quality is studied. Issues concerning the practical derivation of a posteriori estimates for the quality of approximate solutions are addressed, and numerical algorithms for finding such estimates are described. The exposition is illustrated by results of numerical experiments.Beyond the Bakushinkii veto: regularising linear inverse problems without knowing the noise distribution.https://www.zbmath.org/1453.651242021-02-27T13:50:00+00:00"Harrach, Bastian"https://www.zbmath.org/authors/?q=ai:harrach.bastian"Jahn, Tim"https://www.zbmath.org/authors/?q=ai:jahn.tim"Potthast, Roland"https://www.zbmath.org/authors/?q=ai:potthast.roland-w-eSummary: This article deals with the solution of linear ill-posed equations in Hilbert spaces. Often, one only has a corrupted measurement of the right hand side at hand and the Bakushinskii veto tells us, that we are not able to solve the equation if we do not know the noise level. But in applications it is ad hoc unrealistic to know the error of a measurement. In practice, the error of a measurement may often be estimated through averaging of multiple measurements. We integrated that in our anlaysis and obtained convergence to the true solution, with the only assumption that the measurements are unbiased, independent and identically distributed according to an unknown distribution.On the convergence of Lawson methods for semilinear stiff problems.https://www.zbmath.org/1453.652692021-02-27T13:50:00+00:00"Hochbruck, Marlis"https://www.zbmath.org/authors/?q=ai:hochbruck.marlis"Leibold, Jan"https://www.zbmath.org/authors/?q=ai:leibold.jan"Ostermann, Alexander"https://www.zbmath.org/authors/?q=ai:ostermann.alexanderThe convergence behaviour of Lawson-type exponential integrators is studied for semilinear problems. The convergence proof is based on the variation-of-constants formula, whereas the nonlinearity is expanded along with the flow of the homogeneous problem. The expansion of the exact solution is carried out in terms of elementary integrals, and of that of the numerical solution in terms of elementary quadrature rules. It is shown that the non-stiff order conditions together with some assumptions on the exact solution give full order of convergence. The authors prove that a Lawson method converges with order \(p\) if the order of the underlying Runge-Kutta method is at least \(p\) and the solution satisfies appropriate regularity assumptions. The regularity assumptions are related to the corresponding conditions for splitting methods. These conditions are studied for methods of orders one and two, respectively, and is worked out for the nonlinear Schrödinger equation. The error analysis reveals a different behaviour between the first-order Lawson method and
the exponential Euler method, as observed in numerical experiments for the linear Schrödinger equation with periodic boundary conditions.
Reviewer: Bülent Karasözen (Ankara)Finite element error estimates in non-energy norms for the two-dimensional scalar Signorini problem.https://www.zbmath.org/1453.654042021-02-27T13:50:00+00:00"Christof, Constantin"https://www.zbmath.org/authors/?q=ai:christof.constantin"Haubner, Christof"https://www.zbmath.org/authors/?q=ai:haubner.christofSummary: This paper is concerned with error estimates for the piecewise linear finite element approximation of the two-dimensional scalar Signorini problem on a convex polygonal domain \(\varOmega\). Using a Céa-type lemma, a supercloseness result, and a non-standard duality argument, we prove \(W^{1, p}(\varOmega)\)-, \(L^\infty (\varOmega)\)-, \(W^{1, \infty}(\varOmega)\)-, and \(H^{1/2}(\partial \varOmega)\)-error estimates under reasonable assumptions on the regularity of the exact solution and \(L^p(\varOmega)\)-error estimates under comparatively mild assumptions on the involved contact sets. The obtained orders of convergence turn out to be optimal for problems with essentially bounded right-hand sides. Our results are accompanied by numerical experiments which confirm the theoretical findings.A note on Dekker's FastTwoSum algorithm.https://www.zbmath.org/1453.650062021-02-27T13:50:00+00:00"Lange, Marko"https://www.zbmath.org/authors/?q=ai:lange.marko"Oishi, Shin'ichi"https://www.zbmath.org/authors/?q=ai:oishi.shinichiSummary: More than 45 years ago, Dekker proved that it is possible to evaluate the exact error of a floating-point sum with only two additional floating-point operations, provided certain conditions are met. Today the respective algorithm for transforming a sum into its floating-point approximation and the corresponding error is widely referred to as FastTwoSum. Besides some assumptions on the floating-point system itself -- all of which are satisfied by any binary IEEE 754 standard conform arithmetic, the main practical limitation of FastTwoSum is that the summands have to be ordered according to their exponents. In most preceding applications of FastTwoSum, however, a more stringent condition is used, namely that the summands have to be sorted according to their absolute value. In remembrance of Dekker's work, this note reminds the original assumptions for an error-free transformation via FastTwoSum. Moreover, we generalize the conditions for arbitrary bases and discuss a possible modification of the FastTwoSum algorithm to extend its applicability even further. Subsequently, a range of programs exploiting the wider applicability is presented. This comprises the OnlineExactSum algorithm by Zhu and Hayes, an error-free transformation from a product of three floating-point numbers to a sum of the same number of addends, and an algorithm for accurate summation proposed by Demmel and Hida.Convergence of adaptive filtered schemes for first order evolutionary Hamilton-Jacobi equations.https://www.zbmath.org/1453.652142021-02-27T13:50:00+00:00"Falcone, Maurizio"https://www.zbmath.org/authors/?q=ai:falcone.maurizio"Paolucci, Giulio"https://www.zbmath.org/authors/?q=ai:paolucci.giulio"Tozza, Silvia"https://www.zbmath.org/authors/?q=ai:tozza.silviaSummary: We consider a class of ``filtered'' schemes for first order time dependent Hamilton-Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function \(F\) which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter \(\varepsilon =\varepsilon ({\Delta t,\Delta x})>0\) which goes to 0 as the time and space steps \((\Delta t,\Delta x)\) are going to 0 and does not depend on the time \(t_n\), for each iteration \(n\). The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of \(\varepsilon =\varepsilon^n (\Delta t, \Delta x)\) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold \(\varepsilon^n\). A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.A least-squares virtual element method for second-order elliptic problems.https://www.zbmath.org/1453.654192021-02-27T13:50:00+00:00"Wang, Ying"https://www.zbmath.org/authors/?q=ai:wang.ying.8|wang.ying.4|wang.ying|wang.ying.2|wang.ying.3|wang.ying.1|wang.ying.6"Wang, Gang"https://www.zbmath.org/authors/?q=ai:wang.gang.5|wang.gang.3|wang.gang|wang.gang.1|wang.gang.4|wang.gang.2Summary: In this paper, a least-squares virtual element method is presented for approximating the vector and scalar variables of second-order elliptic problems. The \(\boldsymbol{H}(\operatorname{div})\)-conforming and scalar-conforming virtual elements are used to approximate the vector and scalar variables, respectively. The method allows the use of very general polygonal meshes and leads to a symmetric positive definite system. The optimal a priori error estimates are established for the vector variable in \(\boldsymbol{H}(\operatorname{div})\) norm and the scalar variable in \(H^1\) norm. A simple a posteriori error estimator is also presented and proved to be reliable and efficient. The virtual element method handles the hanging nodes naturally, thus the local mesh post-processing to remove hanging nodes is not required. Numerical experiments are conducted to verify the accuracy of the method, and show the effectiveness and flexibility of the adaptive strategy driven by the proposed estimator and suitable mesh refinement strategy.Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling.https://www.zbmath.org/1453.653892021-02-27T13:50:00+00:00"Droniou, Jérôme"https://www.zbmath.org/authors/?q=ai:droniou.jerome"Medla, Matej"https://www.zbmath.org/authors/?q=ai:medla.matej"Mikula, Karol"https://www.zbmath.org/authors/?q=ai:mikula.karolSummary: We develop and analyse finite volume methods for the Poisson problem with boundary conditions involving oblique derivatives. We design a generic framework, for finite volume discretisations of such models, in which internal fluxes are not assumed to have a specific form, but only to satisfy some (usual) coercivity and consistency properties. The oblique boundary conditions are split into a normal component, which directly appears in the flux balance on control volumes touching the domain boundary, and a tangential component which is managed as an advection term on the boundary. This advection term is discretised using a finite volume method based on a centred discretisation (to ensure optimal rates of convergence) and stabilised using a vanishing boundary viscosity. A convergence analysis, based on the 3rd Strang Lemma [\textit{D. A. Di Pietro} and \textit{J. Droniou}, Calcolo 55, No. 3, Paper No. 40, 39 p. (2018; Zbl 06969707)], is conducted in this generic finite volume framework, and yields the expected \(\mathcal{O}(h)\) optimal convergence rate in discrete energy norm. We then describe a specific choice of numerical fluxes, based on a generalised hexahedral meshing of the computational domain. These fluxes are a corrected version of fluxes originally introduced in [\textit{M. Medla} et al., ``Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography'', J. Geod., 92, No. 1, 1--19 (2018; \url{doi:10.1007/s00190-017-1040-z})]. We identify mesh regularity parameters that ensure that these fluxes satisfy the required coercivity and consistency properties. The theoretical rates of convergence are illustrated by an extensive set of 3D numerical tests, including some conducted with two variants of the proposed scheme. A test involving real-world data measuring the disturbing potential in Earth gravity modelling over Slovakia is also presented.Versatile mixed methods for the incompressible Navier-Stokes equations.https://www.zbmath.org/1453.760662021-02-27T13:50:00+00:00"Chen, Xi"https://www.zbmath.org/authors/?q=ai:chen.xi.5|chen.xi.2|chen.xi|chen.xi.1|chen.xi.4"Williams, David M."https://www.zbmath.org/authors/?q=ai:williams.david-mSummary: In the spirit of the ``Principle of equipresence'' introduced by \textit{C. Truesdell} and \textit{R. Toupin} [``The classical field theories'', in: Principles of classical mechanics and field theory. Berlin, Heidelberg: Springer Verlag. 226--858 (1960; \url{doi = {10.1007/978-3-642-45943-6_2}})], we use the full version of the viscous stress tensor \(\nu\left(\nabla\boldsymbol{u}+\nabla\boldsymbol{u}^T-\frac{2}{3}(\nabla\cdot\boldsymbol{u})\mathbb{I}\right)\) which was originally derived for compressible flows, instead of the classical incompressible stress tensor \(\nu\nabla\boldsymbol{u}\). (Note that, here \(\nu\) is the dynamic viscosity coefficient, and \(\boldsymbol{u}\) is the velocity field.) In our approach, the divergence-free constraint for the viscous stress term is not enforced ahead of discretization. Instead, our formulation allows the scheme itself to ``choose'' a consistent way to interpret the divergence-free constraint: i.e., the divergence-free constraint is interpreted (or enforced) in a consistent fashion in both the mass conservation equation \textit{and} the stress tensor term (in the momentum equation). Furthermore, our approach preserves the original symmetrical properties of the stress tensor, e.g. its rotational invariance, and it remains physically correct in the context of compressible flows. As a result, our approach facilitates versatility and code reuse. In this paper, we introduce our approach and establish some important mathematical properties for the resulting class of finite element schemes. More precisely, for general mixed methods, which are not necessarily pointwise divergence-free, we establish the existence of a new norm induced by the full, viscous bilinear form. Thereafter, we prove the coercivity of the viscous bilinear form and the semi-coercivity of a convective trilinear form. In addition, we demonstrate L2-stability of the discrete velocity fields for the general class of methods and (by deduction) the \(H(\mathrm{div})\)-conforming methods. Finally, we run some numerical experiments to illustrate the behavior of the versatile mixed methods, and we make careful comparisons with a conventional \(H(\mathrm{div})\)-conforming scheme.Hyperbolic problems: theory, numerics, applications. Proceedings of the 17th international conference, HYP2018, Pennsylvania State University, University Park, PA, USA, June 25--29, 2018.https://www.zbmath.org/1453.350032021-02-27T13:50:00+00:00"Bressan, Alberto (ed.)"https://www.zbmath.org/authors/?q=ai:bressan.alberto"Lewicka, Marta (ed.)"https://www.zbmath.org/authors/?q=ai:lewicka.marta-wang-dehua"Wang, Dehua (ed.)"https://www.zbmath.org/authors/?q=ai:wang.dehua"Zheng, Yuxi (ed.)"https://www.zbmath.org/authors/?q=ai:zheng.yuxiPublisher's description: This volume contains the proceedings of the XVII international conference (HYP2018) on hyperbolic problems, which was held at the Pennsylvania State University, University Park, on June 25--29, 2018.
The contributions collected in this volume cover a wide range of topics. Some of these represent the latest developments on classical multi-dimensional problems, dealing with shock reflections and withthe stability of vortices and boundary layers. Other contributions provide sharp results on the structure and regularity of solutions toconservation laws, or discuss the fine line between well-posedness and ill-posedness for transport equations with rough coefficients, and for the equations of inviscid fluid flow. Further progress is reported at the interface between hyperbolic and kinetic models, including the hydrodynamic limit of the Boltzmann equation. Kinetic andmacroscopic models for collective dynamics of many-body systems, which have attracted much interest in recent years, are also covered inthis volume. Finally, a large number of papers are devoted to advancesin computational methods, with diverse applications such as: submarine avalanches, tsunami waves, chemically reacting flows, solitary waves,gas flow on a network of pipelines, traffic flow with multiple types ofvehicles, etc.
The present volume provides a timely survey of the state of the art, which will be of interest to researchers, students and practitioners, withinterest in the theoretical, computational and applied aspects of hyperbolic problems.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Chen, Gui-Qiang G.; Feldman, Mikhail; Xiang, Wei}, Uniqueness and stability for the shock reflection-diffraction problem for potential flow, 2-24 [Zbl 07315450]
\textit{Chertock, Alina; Kurganov, Alexander; Miller, Jason; Yan, Jun}, Central-upwind scheme for a non-hydrostatic Saint-Venant system, 25-41 [Zbl 07315451]
\textit{Gallay, Thierry}, Stability of vortices in ideal fluids: the legacy of Kelvin and Rayleigh, 42-59 [Zbl 07315452]
\textit{Guo, Yan}, On the Euler-Poisson system, 60-75 [Zbl 07315453]
\textit{Huang, Feimin}, The hydrodynamic limit of the Boltzmann equation for Riemann solutions, 76-97 [Zbl 07315454]
\textit{Wang, Xiang; Wang, Ya-Guang}, Well-posedness of boundary layer problem in wind-driven oceanic circulation, 98-111 [Zbl 07315455]
\textit{Alonso, Ricardo}, On the dynamic of dissipative particles, 113-123 [Zbl 07315456]
\textit{Bae, Myoungjean; Xiang, Wei}, A note on 2-D detached shocks of steady Euler system, 124-135 [Zbl 07315457]
\textit{Carles, Rémi; Carrapatoso, Kleber; Hillairet, Matthieu}, Rigidity in generalized isothermal fluids, 136-144 [Zbl 07315458]
\textit{Choi, Young-Pil; Jung, Jinwook}, Asymptotic analysis for Vlasov-Fokker-Planck/compressible Navier-Stokes equations with a density-dependent viscosity, 145-163 [Zbl 07315459]
\textit{Daneri, Sara; Runa, Eris}, On non-uniqueness below Onsager's critical exponent, 164-174 [Zbl 07315460]
\textit{Fernández-Nieto, Enrique D.; Castro-Dıaz, Manuel J.; Mangeney, Anne}, Modelling, numerical method and analysis of the collapse of cylindrical submarines granular mass, 175-191 [Zbl 07315461]
\textit{Kagei, Yoshiyuki}, On stationary bifurcation problem for the compressible Navier-Stokes equations, 192-202 [Zbl 07315462]
\textit{Liu, Hailiang}, On structure-preserving high order methods for conversation laws, 203-213 [Zbl 07315463]
\textit{Abgrall, Rémi; le Mélédo, Elise; Öffner, Philipp; Ranocha, Hendrik}, Error boundedness of correction procedure via reconstruction / flux reconstruction and the connection to residual distribution schemes, 215-222 [Zbl 07315464]
\textit{Abreu, Eduardo; Lambert, Wanderson; Pérez, John; Santo, Arthur}, A weak asymptotic solution analysis for a Lagrangian-Eulerian scheme for scalar hyperbolic conservation laws, 223-230 [Zbl 07315465]
\textit{Amadori, Debora; Aqel, Al-Zahra' Fatima; Dal Santo, Edda}, Decay in \(L^\infty\) for the damped semilinear wave equation on a bounded 1d domain, 231-238 [Zbl 07315466]
\textit{Ancona, Fabio; Caravenna, Laura; Christoforou, Cleopatra}, On \(L^1\)-stability of BV solutions for a model of granular flow, 239-247 [Zbl 07315467]
\textit{Ancona, Fabio; Glass, Olivier; Nguyen, Khai T.}, Quantitative compactness estimate for scalar conservation laws with non-convex fluxes, 248-255 [Zbl 07315468]
\textit{Antonelli, Paolo; Hientzsch, Lars Eric; Marcati, Pierangelo}, The incompressible limit for finite energy weak solutions of quantum Navier-Stokes equations, 256-263 [Zbl 07315469]
\textit{Antonelli, Paolo; Marcati, Pierangelo; Zheng, Hao}, 1D quantum hydrodynamic system: global existence, stability and dispersion, 264-270 [Zbl 07315470]
\textit{Aregba-Driollet, Denise; Brull, Stéphane}, About viscous approximations of the bitemperature Euler system, 271-278 [Zbl 07315471]
\textit{Arun, K. R.; Samantaray, Saurav}, An asymptotic preserving time integrator for low Mach number limits of the Euler equations with gravity, 279-286 [Zbl 07315472]
\textit{Baranger, Céline; Bisi, Marzia; Brull, Stéphane; Desvillettes, Laurent}, On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases, 287-295 [Zbl 07315473]
\textit{Barsukow, Wasilij}, Stationary states of finite volume discretizations of multi-dimensional linear hyperbolic systems, 296-303 [Zbl 07315474]
\textit{Berjamin, Harold; Junca, Stéphane; Lombard, Bruno}, Smooth solutions for nonlinear elastic waves with softening, 304-311 [Zbl 07315475]
\textit{Bianchini, Stefano; Bonicatto, Paolo}, Untangling of trajectories for non-smooth vector fields and Bressan's compactness conjecture, 312-327 [Zbl 07315476]
\textit{Bressan, Alberto; Guerra, Graziano; Shen, Wen}, Conservation laws with regulated fluxes, 328-335 [Zbl 07315477]
\textit{Burtscher, Annegret Y.}, Initial data and black holes for matter models, 336-345 [Zbl 07315478]
\textit{Cacciafesta, Federico}, Dispersive dynamics of the Dirac equation on curved spaces, 346-352 [Zbl 07315479]
\textit{Chiarello, Felisia A.; Goatin, Paola; Villada, Luis M.}, High-order finite volume WENO schemes for non-local multi-class traffic flow models, 353-360 [Zbl 07315480]
\textit{Ciampa, Gennaro; Crippa, Gianluca; Spirito, Stefano}, On smooth approximations of rough vector fields and the selection of flows, 361-368 [Zbl 07315481]
\textit{Colombo, Maria; Crippa, Gianluca; Graff, Marie; Spinolo, Laura V.}, Recent results on the singular local limit for nonlocal conservation laws, 369-376 [Zbl 07315482]
\textit{Colombo, Rinaldo M.; Garavello, Mauro}, A feedback strategy in hyperbolic control problems, 377-384 [Zbl 07315483]
\textit{Coquel, Frédéric; Marmignon, Claude; Rai, Pratik; Renac, Florent}, Adjoint approximation of nonlinear hyperbolic systems with non-conservative products, 385-392 [Zbl 07315484]
\textit{Corli, Andrea; Malaguti, Luisa}, Models of collective movements with negative degenerate diffusivities, 393-399 [Zbl 07315485]
\textit{Courtès, Clémentine; Franck, Emmanuel}, Linear stability of a vectorial kinetic relaxation scheme with a central velocity, 400-407 [Zbl 07315486]
\textit{Dond, Asha K.; Gudi, Thirupathi}, A posteriori error analysis for patch-wise local projection stabilized FEM for convection-diffusion problems, 408-418 [Zbl 07315487]
\textit{Dymski, Nikodem; Goatin, Paola; Rosini, Massimiliano D.}, Modeling moving bottlenecks on road networks, 419-426 [Zbl 07315488]
\textit{Egger, Herbert; Kugler, Thomas; Liljegren-Sailer, Björn}, Stability preserving approximations of a semilinear hyperbolic gas transport model, 427-433 [Zbl 07315489]
\textit{Folino, Raffaele; Lattanzio, Corrado; Mascia, Corrado}, Motion of interfaces for hyperbolic variations of the Allen-Cahn equation, 434-441 [Zbl 07315490]
\textit{Giesselmann, Jan; Joshi, Hrishikesh}, Model adaptation of chemically reacting flows based on a posteriori error estimates, 442-448 [Zbl 07315491]
\textit{Giesselmann, Jan; Meyer, Fabian; Rohde, Christian}, An a posteriori error analysis based on non-intrusive spectral projections for systems of random conservation laws, 449-456 [Zbl 07315492]
\textit{Gong, Xiaoqian; Kawski, Matthias}, Analysis of a nonlinear hyperbolic conservation law with measure-valued data, 457-464 [Zbl 07315493]
\textit{Jagtap, Ameya D.}, Higher order scheme for sine-Gordon equation in nonlinear non-homogeneous media, 465-474 [Zbl 07315494]
\textit{Keimer, Alexander; Pflug, Lukas; Spinola, Michele}, Nonlocal balance laws. Results on existence, uniqueness and regularity, 475-482 [Zbl 07315495]
\textit{Keller, Laura Gioia Andrea}, Homogenization with two kinds of microstructures: from the microscopic to the macroscopic description of concentrations of chemical agents, 483-490 [Zbl 07315496]
\textit{Klingenberg, Christian; Markfelder, Simon}, Non-uniqueness of entropy-conserving solutions to the ideal compressible MHD equations, 491-498 [Zbl 07315497]
\textit{Knapp, Stephan}, Piecewise deterministic Markov processes driven by scalar conservation laws, 499-506 [Zbl 07315498]
\textit{Kroener, Dietmar; Mackeben, Thomas; Rokyta, Mirko}, Nonconvergence proof for the LDA-scheme, 507-514 [Zbl 07315499]
\textit{Kumar, Rakesh}, Hybrid FDM-WENO method for the convection-diffusion problems, 515-523 [Zbl 07315500]
\textit{Liard, Thibault; Marcellini, Francesca; Piccoli, Benedetto}, The Riemann problem for the GARZ model with a moving constraint, 524-530 [Zbl 07315501]
\textit{Makino, Tetu}, Recent progress of the study of hydrodynamic evolution of gaseous stars, 531-537 [Zbl 07315502]
\textit{Mantri, Yogiraj; Herty, Michael; Noelle, Sebastian}, Well-balanced scheme for network of gas pipelines, 538-545 [Zbl 07315503]
\textit{Marconi, Elio}, Structure and regularity of solutions to 1d scalar conservation laws, 546-553 [Zbl 07315504]
\textit{Marroquin, Daniel R.}, Recent progress on the study of the short wave-Long wave interactions system for aurora-type phenomena, 554-561 [Zbl 07315505]
\textit{Modena, Stefano}, On some recent results concerning non-uniqueness for the transport equation, 562-568 [Zbl 07315506]
\textit{Morando, Alessandro; Trebeschi, Paola; Wang, Tao}, Existence and stability of nonisentropic compressible vortex sheets, 569-576 [Zbl 07315507]
\textit{Nedeljkov, Marko; Neumann, Lukas; Oberguggenberger, Michael}, Spherically symmetric shadow wave solutions to the compressible Euler system at the origin, 577-585 [Zbl 07315508]
\textit{Ostrowski, Lukas; Rohde, Christian}, Phase field modelling for compressible droplet impingement, 586-593 [Zbl 07315509]
\textit{Pelanti, Marica}, A Roe-like reformulation of the HLLC Riemann solver and applications, 594-602 [Zbl 07315510]
\textit{Pichard, Teddy}, Existence of steady two-phase flows with discontinuous boiling effects, 603-610 [Zbl 07315511]
\textit{Prigent, Corentin; Brull, Stephane; Dubroca, Bruno}, A kinetic approach to the bi-temperature Euler model, 611-620 [Zbl 07315512]
\textit{Rugamba, Jean; Zeng, Yanni}, Pointwise asymptotic behavior of a chemotaxis model, 621-629 [Zbl 07315513]
\textit{Shu, Jingyang}, Fronts for the SQG equation: a review, 630-638 [Zbl 07315514]
\textit{Srivastava, Varsha}, Robust numerical method for time-dependent singularly perturbed semilinear problems, 639-648 [Zbl 07315515]
\textit{Strani, Marta}, The role of a regularization in hyperbolic instabilities, 649-657 [Zbl 07315516]
\textit{Suzuki, Masahiro}, On the Degond-Lucquin-Desreux-Morrow model for gas discharge, 658-665 [Zbl 07315517]
\textit{Tsuge, Naoki}, Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow, 666-673 [Zbl 07315518]
\textit{Wang, Yong}, On a class of new generalized Poisson-Nernst-Planck-Navier-Stokes equations, 674-681 [Zbl 07315519]
\textit{Giesselmann, Jan; Zacharenakis, Dimitrios}, A posteriori analysis for the Navier-Stokes-Korteweg model, 682-690 [Zbl 07315520]Finite difference/finite element method for tempered time fractional advection-dispersion equation with fast evaluation of Caputo derivative.https://www.zbmath.org/1453.653112021-02-27T13:50:00+00:00"Cao, Jiliang"https://www.zbmath.org/authors/?q=ai:cao.jiliang"Xiao, Aiguo"https://www.zbmath.org/authors/?q=ai:xiao.aiguo"Bu, Weiping"https://www.zbmath.org/authors/?q=ai:bu.weipingSummary: In this paper, a class of fractional advection-dispersion equations with Caputo tempered fractional derivative are considered numerically. An efficient algorithm for the evaluation of Caputo tempered fractional derivative is proposed to sharply reduce the computational work and storage, and this is of great significance for large-scale problems. Based on the nonsmooth regularity assumptions, a semi-discrete form is obtained by finite difference method in time, and its stability and convergence are investigated. Then by finite element method, we derive the corresponding fully discrete scheme and discuss its convergence. At last, some numerical examples, based on different domains, are presented to demonstrate effectiveness of numerical schemes and confirm the theoretical analysis.Imprecise Monte Carlo simulation and iterative importance sampling for the estimation of lower previsions.https://www.zbmath.org/1453.600062021-02-27T13:50:00+00:00"Troffaes, Matthias C. M."https://www.zbmath.org/authors/?q=ai:troffaes.matthias-c-mSummary: We develop a theoretical framework for studying numerical estimation of lower previsions, generally applicable to two-level Monte Carlo methods, importance sampling methods, and a wide range of other sampling methods one might devise. We link consistency of these estimators to Glivenko-Cantelli classes, and for the sub-Gaussian case we show how the correlation structure of this process can be used to bound the bias and prove consistency. We also propose a new upper estimator, which can be used along with the standard lower estimator, in order to provide a simple confidence interval. As a case study of this framework, we then discuss how importance sampling can be exploited to provide accurate numerical estimates of lower previsions. We propose an iterative importance sampling method to drastically improve the performance of imprecise importance sampling. We demonstrate our results on the imprecise Dirichlet model.Dual-criteria time stepping for weakly compressible smoothed particle hydrodynamics.https://www.zbmath.org/1453.761722021-02-27T13:50:00+00:00"Zhang, Chi"https://www.zbmath.org/authors/?q=ai:zhang.chi"Rezavand, Massoud"https://www.zbmath.org/authors/?q=ai:rezavand.massoud"Hu, Xiangyu"https://www.zbmath.org/authors/?q=ai:hu.xiangyuSummary: Implementing particle-interaction configuration, which consists of determining particle-neighbor lists and computing corresponding kernel weights and gradients, and time integration are performance intensive essentials of smoothed particle hydrodynamic (SPH) method. In this paper, a dual-criteria time-stepping method is proposed to improve the computational efficiency of the weakly-compressible SPH (WCSPH) method for modeling incompressible flows. The key idea is to introduce an advection time-step criterion, which is based on flow speed, for recreating the particle-interaction configuration. Within an advection time-step, several steps of pressure relaxation, i.e. the time integration of the particle density, position and velocity due to the action of pressure gradient, according to the acoustic time-step criterion based on the artificial speed of sound, can be carried out without updating the particle-interaction configuration and with much larger time steps compared with the conventional counterpart. The method has shown optimized computational performance through CPU cost analysis. Good accuracy is also obtained for the presented benchmarks implying promising potential of the proposed method for simulating incompressible flows and fluid-structure interactions.Sufficient conditions for a multidimensional system of periodic wavelets to be a frame.https://www.zbmath.org/1453.420322021-02-27T13:50:00+00:00"Andrianov, P. A."https://www.zbmath.org/authors/?q=ai:andrianov.p-aA multivariate periodic wavelet system with matrix dilation is studied in this paper. A sufficient condition for such a system to be a Bessel sequence is provided in terms of the Fourier coefficients of the functions that generate the system. As a consequence the author provides a construction of a pair of biorthogonal dual wavelet bases from a suitable trigonometric polynomial.
Reviewer: Ghanshyam Bhatt (Nashville)High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters.https://www.zbmath.org/1453.653512021-02-27T13:50:00+00:00"Zhu, Jun"https://www.zbmath.org/authors/?q=ai:zhu.jun"Qiu, Jianxian"https://www.zbmath.org/authors/?q=ai:qiu.jianxian"Shu, Chi-Wang"https://www.zbmath.org/authors/?q=ai:shu.chi-wangSummary: In this paper, a new type of multi-resolution weighted essentially non-oscillatory (WENO) limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods is designed. This type of multi-resolution WENO limiters is an extension of the multi-resolution WENO finite volume and finite difference schemes developed in [the first and third author, ibid. 375, 659--683 (2018; Zbl 1416.65286)]. Such new limiters use information of the DG solution essentially only within the troubled cell itself, to build a sequence of hierarchical \(L^2\) projection polynomials from zeroth degree to the highest degree of the RKDG method. The second-order, third-order, fourth-order, and fifth-order RKDG methods with these multi-resolution WENO limiters have been developed as examples, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO limiters can be any positive numbers on the condition that their sum equals one. This is the first time that a series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original DG methods. Benchmark examples are given to demonstrate the good performance of these RKDG methods with the associated multi-resolution WENO limiters.Inverse problems. Basics, theory and applications in geophysics. 2nd updated edition.https://www.zbmath.org/1453.650022021-02-27T13:50:00+00:00"Richter, Mathias"https://www.zbmath.org/authors/?q=ai:richter.mathiasPublisher's description: This textbook is an introduction to the subject of inverse problems with an emphasis on practical solution methods and applications from geophysics. The treatment is mathematically rigorous, relying on calculus and linear algebra only; familiarity with more advanced mathematical theories like functional analysis is not required. Containing up-to-date methods, this book will provide readers with the tools necessary to compute regularized solutions of inverse problems. A variety of practical examples from geophysics are used to motivate the presentation of abstract mathematical ideas, thus assuring an accessible approach.Beginning with four examples of inverse problems, the opening chapter establishes core concepts, such as formalizing these problems as equations in vector spaces and addressing the key issue of ill-posedness. Chapter Two then moves on to the discretization of inverse problems, which is a prerequisite for solving them on computers. Readers will be well-prepared for the final chapters that present regularized solutions of inverse problems in finite-dimensional spaces, with Chapter Three covering linear problems and Chapter Four studying nonlinear problems. Model problems reflecting scenarios of practical interest in the geosciences, such as inverse gravimetry and full waveform inversion, are fully worked out throughout the book. They are used as test cases to illustrate all single steps of solving inverse problems, up to numerical computations. Five appendices include the mathematical foundations needed to fully understand the material.This second edition expands upon the first, particularly regarding its up-to-date treatment of nonlinear problems. Following the author's approach, readers will understand the relevant theory and methodology needed to pursue more complex applications. \textit{Inverse Problems} is ideal for graduate students and researchers interested in geophysics and geosciences.
See the review of the first edition in [Zbl 1365.65157].Linearized compact difference methods combined with Richardson extrapolation for nonlinear delay Sobolev equations.https://www.zbmath.org/1453.652392021-02-27T13:50:00+00:00"Zhang, Chengjian"https://www.zbmath.org/authors/?q=ai:zhang.chengjian"Tan, Zengqiang"https://www.zbmath.org/authors/?q=ai:tan.zengqiangSummary: Delay Sobolev equations (DSEs) are a class of important models in fluid mechanics, thermodynamics and the other related fields. For solving this class of equations, in this paper, linearized compact difference methods (LCDMs) for one- and two-dimensional problems of DSEs are suggested. The solvability and convergence of the methods are analyzed and it is proved under some appropriate conditions that the methods are convergent of order two in time and order four in space. In order to improve the computational accuracy of LCDMs in time, we introduce the Richardson extrapolation technique, which leads to the improved LCDMs can reach the fourth-order accuracy in both time and space. Finally, with several numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.Constraint consensus based artificial bee colony algorithm for constrained optimization problems.https://www.zbmath.org/1453.901672021-02-27T13:50:00+00:00"Sun, Liling"https://www.zbmath.org/authors/?q=ai:sun.liling"Wu, Yuhan"https://www.zbmath.org/authors/?q=ai:wu.yuhan"Liang, Xiaodan"https://www.zbmath.org/authors/?q=ai:liang.xiaodan"He, Maowei"https://www.zbmath.org/authors/?q=ai:he.maowei"Chen, Hanning"https://www.zbmath.org/authors/?q=ai:chen.hanningSummary: Over the last few decades, evolutionary algorithms (EAs) have been widely adopted to solve complex optimization problems. However, EAs are powerless to challenge the constrained optimization problems (COPs) because they do not directly act to reduce constraint violations of constrained problems. In this paper, the robustly global optimization advantage of artificial bee colony (ABC) algorithm and the stably minor calculation characteristic of constraint consensus (CC) strategy for COPs are integrated into a novel hybrid heuristic algorithm, named ABCCC. CC strategy is fairly effective to rapidly reduce the constraint violations during the evolutionary search process. The performance of the proposed ABCCC is verified by a set of constrained benchmark problems comparing with two state-of-the-art CC-based EAs, including particle swarm optimization based on CC (PSOCC) and differential evolution based on CC (DECC). Experimental results demonstrate the promising performance of the proposed algorithm, in terms of both optimization quality and convergence speed.Numerical computations of split Bregman method for fourth order total variation flow.https://www.zbmath.org/1453.651422021-02-27T13:50:00+00:00"Giga, Yoshikazu"https://www.zbmath.org/authors/?q=ai:giga.yoshikazu"Ueda, Yuki"https://www.zbmath.org/authors/?q=ai:ueda.yukiSummary: The split Bregman framework for Osher-Solé-Vese (OSV) model and fourth order total variation flow are studied. We discretize the problem by piecewise constant function and compute \(\nabla (-\Delta_{\operatorname{av}})^{-1}\) approximately and exactly. Furthermore, we provide a new shrinkage operator for Spohn's fourth order model. Numerical experiments are demonstrated for fourth order problems under periodic boundary condition.A new high-order numerical method for solving singular two-point boundary value problems.https://www.zbmath.org/1453.651752021-02-27T13:50:00+00:00"Roul, Pradip"https://www.zbmath.org/authors/?q=ai:roul.pradip"Thula, Kiran"https://www.zbmath.org/authors/?q=ai:thula.kiranSummary: In [Comput. Math. Appl. 64, No. 2, 115--120 (2012; Zbl 1252.65134)], \textit{J. Goh} et al. proposed a numerical technique based on quartic B-spline collocation for solving a class of singular boundary value problems (SBVP) with Neumann and Dirichlet boundary conditions (BC). This method is only fourth-order accurate. In this paper, we propose an optimal numerical technique for solving a more general class of nonlinear SBVP subject to Neumann and Robin BC. The method is based on high order perturbation of the problem under consideration. The convergence of the proposed method is analyzed. To demonstrate the applicability and efficiency of the method, we consider four numerical examples, three of which arise in various physical models in applied science and engineering. A comparison with other available numerical solutions has been carried out to justify the advantage of the proposed technique. Numerical result reveals that the proposed method is sixth order convergent, which in turn is two orders of magnitude larger than in [loc. cit.].A multiresolution algorithm to approximate the Hutchinson measure for IFS and GIFS.https://www.zbmath.org/1453.280052021-02-27T13:50:00+00:00"da Cunha, Rudnei D."https://www.zbmath.org/authors/?q=ai:da-cunha.rudnei-d"Oliveira, Elismar R."https://www.zbmath.org/authors/?q=ai:oliveira.elismar-r"Strobin, Filip"https://www.zbmath.org/authors/?q=ai:strobin.filipSummary: We introduce a discrete version of the Hutchinson-Barnsley theory providing algorithms to approximate the Hutchinson measure for iterated function systems (IFS) and generalized iterated function systems (GIFS), complementing the discrete version of the deterministic algorithm considered in our previous work.Multiscale texture orientation analysis using spectral total-variation decomposition.https://www.zbmath.org/1453.940092021-02-27T13:50:00+00:00"Horesh, Dikla"https://www.zbmath.org/authors/?q=ai:horesh.dikla"Gilboa, Guy"https://www.zbmath.org/authors/?q=ai:gilboa.guySummary: Multi-level texture separation can considerably improve texture analysis, a significant component in many computer vision tasks. This paper aims at obtaining precise local texture orientations of images in a multiscale manner, characterizing the main obvious ones as well as the very subtle ones. We use the total variation spectral framework to decompose the image into its different textural scales. Gabor filter banks are then employed to detect prominent orientations within the multiscale representation. A necessary condition for perfect texture separation is given, based on the spectral total-variation theory. We show that using this method we can detect and differentiate a mixture of overlapping textures and obtain with high fidelity a multi-valued orientation representation of the image.
For the entire collection see [Zbl 1362.68008].A two-way coupled Euler-Lagrange method for simulating multiphase flows with discontinuous Galerkin schemes on arbitrary curved elements.https://www.zbmath.org/1453.760672021-02-27T13:50:00+00:00"Ching, Eric J."https://www.zbmath.org/authors/?q=ai:ching.eric-j"Brill, Steven R."https://www.zbmath.org/authors/?q=ai:brill.steven-r"Barnhardt, Michael"https://www.zbmath.org/authors/?q=ai:barnhardt.michael"Ihme, Matthias"https://www.zbmath.org/authors/?q=ai:ihme.matthiasSummary: In this work, we develop a Lagrangian point-particle method to support high-speed dusty flow simulations with discontinuous Galerkin schemes. The carrier fluid is treated in an Eulerian frame through the solution of the compressible Navier-Stokes equations. Particle search and localization is based on the geometric mapping of mesh elements to a reference element and is applicable to arbitrary unstructured, curved, multidimensional grids. High-order interpolation is used to calculate the gas state at a given particle position, and the back-coupling of particles to the carrier fluid is carried out via a simple, effective procedure. Furthermore, we discuss difficulties associated with accounting for particle-wall collisions on curved, high-aspect-ratio elements. We develop a methodology that appropriately handles such collisions and accurately computes post-collision particle trajectories. We first apply the Euler-Lagrange method to one-way coupled tests and show the benefit of using curved instead of straight-sided elements for dealing with particle-wall collisions. We proceed by considering more complex multiphase test cases with two-way coupling, namely dusty flows over a flat plate and through a converging-diverging nozzle. Our final test case consists of hypersonic dusty flow over a sphere in which the target quantity is dust-induced surface heating augmentation at the stagnation point. Quantitative comparisons with experiments are provided.Müntz pseudo-spectral method: theory and numerical experiments.https://www.zbmath.org/1453.653582021-02-27T13:50:00+00:00"Khosravian-Arab, Hassan"https://www.zbmath.org/authors/?q=ai:khosravian-arab.hassan"Eslahchi, M. R."https://www.zbmath.org/authors/?q=ai:eslahchi.mohammad-rezaSummary: This paper presents two new non-classical Lagrange basis functions which are based on the new Jacobi-Müntz functions presented by the authors recently. These basis functions are, in fact, generalized forms of the newly generated Jacobi-based functions. With respect to these non-classical Lagrange basis functions, two non-classical interpolants are introduced and their error bounds are proved in detail. The pseudo-spectral differentiation (and integration) matrices have been extracted in two different manners. Some numerical experiments are provided to show the efficiency and capability of these newly generated non-classical Lagrange basis functions.Suppression of chaos by incommensurate excitations: theory and experimental confirmations.https://www.zbmath.org/1453.370732021-02-27T13:50:00+00:00"Martínez, Pedro J."https://www.zbmath.org/authors/?q=ai:martinez.pedro-j"Euzzor, Stefano"https://www.zbmath.org/authors/?q=ai:euzzor.stefano"Meucci, Riccardo"https://www.zbmath.org/authors/?q=ai:meucci.riccardo"Chacón, Ricardo"https://www.zbmath.org/authors/?q=ai:chacon.ricardoSummary: We experimentally, numerically, and theoretically characterize the effectiveness of incommensurate excitations at suppressing chaos in damped driven systems. Specifically, we consider an inertial Brownian particle moving in a prototypical two-well potential and subjected to a primary (chaos-inducing) harmonic excitation and a suppressory incommensurategeneric (non-harmonic) excitation. We show that the effective amplitude of the suppressory excitation is minimal when the impulse transmitted by it is near its maximum, while its value is rather insensitive to higher-order convergents of the irrational ratio between the involved driving periods. Remarkably, the number and values of the effective initial phase difference between the two excitations are independent of the impulse while they critically depend on each particular convergent in a complex way involving both the approximate frustration of chaos-inducing homoclinic bifurcations and the maximum survival of relevant spatio-temporal symmetries of the dynamical equation.Higher order nonuniform grids for singularly perturbed convection-diffusion-reaction problems.https://www.zbmath.org/1453.651802021-02-27T13:50:00+00:00"Sehar Iqbal"https://www.zbmath.org/authors/?q=ai:sehar-iqbal."Zegeling, Paul Andries"https://www.zbmath.org/authors/?q=ai:zegeling.paul-andriesSummary: In this paper, a higher order nonuniform grid strategy is developed for solving singularly perturbed convection-diffusion-reaction problems with boundary layers. A new nonuniform grid finite difference method (FDM) based on a coordinate transformation is adopted to establish higher order accuracy. To achieve this, we study and make use of the truncation error of the discretized system to obtain a \textit{fourth}-order nonuniform grid transformation. Considering a three-point central finite-difference scheme, we create not only \textit{fourth}-order but even \textit{sixth}-order approximations (which is the maximum order that can be obtained) by a suitable choice of the underlying nonuniform grids. Further, an adaptive nonuniform grid method based on equidistribution principle is used to demonstrate the \textit{sixth}-order of convergence. Unlike several other adaptive numerical methods, our strategy uses no pre-knowledge of the location and the width of the layers. Numerical experiments for various test problems are presented to verify the theoretical aspects. We also show that other, slightly different, choices of the grid distributions already lead to a substantial degradation of the accuracy. The numerical results illustrate the effectiveness of the proposed higher order numerical strategy for nonlinear convection dominated singularly perturbed boundary value problems.The Sinkhorn algorithm, parabolic optimal transport and geometric Monge-Ampère equations.https://www.zbmath.org/1453.350842021-02-27T13:50:00+00:00"Berman, Robert J."https://www.zbmath.org/authors/?q=ai:berman.robert-jAuthor's abstract: We show that the discrete Sinkhorn algorithm -- as applied in the setting of Optimal Transport on a compact manifold -- converges to the solution of a fully non-linear parabolic PDE of Monge-Ampère type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited.
Reviewer: Dian K. Palagachev (Bari)On variational and PDE-based methods for accurate distance function estimation.https://www.zbmath.org/1453.650422021-02-27T13:50:00+00:00"Fayolle, P.-A."https://www.zbmath.org/authors/?q=ai:fayolle.pierre-alain"Belyaev, A. G."https://www.zbmath.org/authors/?q=ai:belyaev.alexander-gSummary: A new variational problem for accurate approximation of the distance from the boundary of a domain is proposed and studied. It is shown that the problem can be efficiently solved by the alternating direction method of multipliers. Links between this problem and \(p\)-Laplacian diffusion are established and studied. Advantages of the proposed distance function estimation method are demonstrated by numerical experiments.Why do we need Voronoi cells and Delaunay meshes? Essential properties of the Voronoi finite volume method.https://www.zbmath.org/1453.652512021-02-27T13:50:00+00:00"Gärtner, K."https://www.zbmath.org/authors/?q=ai:gartner.klaus"Kamenski, L."https://www.zbmath.org/authors/?q=ai:kamenski.lennardAuthors' abstract: Unlike other schemes that locally violate the essential stability properties of the analytic parabolic and elliptic problems, Voronoi finite volume methods (FVM) and boundary conforming Delaunay meshes provide good approximation of the geometry of a problem and are able to preserve the essential qualitative properties of the solution for any given resolution in space and time as well as changes in time scales of multiple orders of magnitude. This work provides a brief description of the essential and useful properties of the Voronoi FVM, application examples, and a motivation why Voronoi FVM deserve to be used more often in practice than they are currently.
Reviewer: Victor Michel-Dansac (Strasbourg)Acceleration of the scheduled relaxation Jacobi method: promising strategies for solving large, sparse linear systems.https://www.zbmath.org/1453.650722021-02-27T13:50:00+00:00"Kong, Qian"https://www.zbmath.org/authors/?q=ai:kong.qian"Jing, Yan-Fei"https://www.zbmath.org/authors/?q=ai:jing.yanfei"Huang, Ting-Zhu"https://www.zbmath.org/authors/?q=ai:huang.ting-zhu"An, Heng-Bin"https://www.zbmath.org/authors/?q=ai:an.hengbinSummary: The main aim of this paper is to develop two algorithms based on the Scheduled Relaxation Jacobi (SRJ) method [\textit{X. I. A. Yang} and \textit{R. Mittal}, ibid. 274, 695--708 (2014; Zbl 1352.65113)] for solving problems arising from the finite-difference discretization of elliptic partial differential equations on large grids. These two algorithms are the Alternating Anderson-Scheduled Relaxation Jacobi (AASRJ) method by utilizing Anderson mixing after each SRJ iteration cycle and the Minimal Residual Scheduled Relaxation Jacobi (MRSRJ) method by minimizing residual after each SRJ iteration cycle, respectively. Through numerical experiments, we show that AASRJ is competitive with the optimal version of the SRJ method [\textit{J. E. Adsuara} et al., ibid. 332, 446--460 (2017; Zbl 1378.65081)] in most problems we considered here, and MRSRJ outperforms SRJ in all cases. The properties of AASRJ and MRSRJ are demonstrated. Both of them are promising strategies for solving large, sparse linear systems while maintaining the simplicity of the Jacobi method.A data assimilation process for linear ill-posed problems.https://www.zbmath.org/1453.651282021-02-27T13:50:00+00:00"Yang, X.-M."https://www.zbmath.org/authors/?q=ai:yang.ximing|yang.xumin|yang.xiumeng|yang.xiaoming|yang.xuemei|yang.xinming|yang.xiaomin|yang.xuemin|yang.xiumin|yang.xingmin|yang.xingming|yang.xueming|yang.xiaomei|yang.ximei|yang.xiaomeng|yang.xinmai|yang.xiumei|yang.xiuming|yang.xinmin.1|yang.xuemeng"Deng, Z.-L."https://www.zbmath.org/authors/?q=ai:deng.zili|deng.zilong|deng.zelin|deng.zhilong|deng.zhenglong|deng.zhili|deng.zhiliangSummary: In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method.A family of convolution-based generalized Stockwell transforms.https://www.zbmath.org/1453.420072021-02-27T13:50:00+00:00"Srivastava, H. M."https://www.zbmath.org/authors/?q=ai:srivastava.hari-mohan"Shah, Firdous A."https://www.zbmath.org/authors/?q=ai:shah.firdous-ahmad"Tantary, Azhar Y."https://www.zbmath.org/authors/?q=ai:tantary.azhar-ySummary: The main purpose of this paper is to introduce a family of convolution-based generalized Stockwell transforms in the context of time-fractional-frequency analysis. The spirit of this article is completely different from two existing studies (see \textit{D. P. Xu} and \textit{K. Guo} [``Fractional \(S\)-transform-Part 1: Theory'', Appl. Geophys. 9, 73--79 (2012)] and \textit{S. K. Singh} [J. Pseudo-Differ. Oper. Appl. 4, No. 2, 251--265 (2013; Zbl 1272.42025)]) in the sense that our approach completely relies on the convolution structure associated with the fractional Fourier transform. We first study all of the fundamental properties of the generalized Stockwell transform, including a relationship between the fractional Wigner distribution and the proposed transform. In the sequel, we introduce both the semi-discrete and discrete counterparts of the proposed transform. We culminate our investigation by establishing some Heisenberg-type inequalities for the generalized Stockwell transform in the fractional Fourier domain.Solving differential equations by wavelet transform method based on the Mother wavelets \& differential invariants.https://www.zbmath.org/1453.420352021-02-27T13:50:00+00:00"Yazdani, Hamid Reza"https://www.zbmath.org/authors/?q=ai:yazdani.hamid-reza"Nadjafikhah, Mehdi"https://www.zbmath.org/authors/?q=ai:nadjafikhah.mehdi"Toomanian, Megerdich"https://www.zbmath.org/authors/?q=ai:toomanian.megerdichSummary: Nowadays, wavelets have been widely used in various fields of science and technology. Meanwhile, the wavelet transforms and the generation of new Mother wavelets are noteworthy. In this paper, we generate new Mother wavelets and analyze the differential equations by using of their corresponding wavelet transforms. This method by Mother wavelets and the corresponding wavelet transforms produces analytical solutions for PDEs and ODEs.Numerical solution to a control problem for integro-differential equations.https://www.zbmath.org/1453.654452021-02-27T13:50:00+00:00"Assanova, A. T."https://www.zbmath.org/authors/?q=ai:assanova.anar-turmaganbetkyzy"Bakirova, E. A."https://www.zbmath.org/authors/?q=ai:bakirova.elmira-a"Kadirbayeva, Zh. M."https://www.zbmath.org/authors/?q=ai:kadirbayeva.zhazira-m|kadirbaeva.zh-mSummary: A control problem for an integro-differential equation is approximated by a problem with parameter for a loaded differential equation. A mutual relationship between the qualitative properties of the original and approximate problems is obtained, and estimates for the differences between their solutions are given. A numerical-approximate method for solving the control problem for the integro-differential equation is proposed, and the convergence, stability, and accuracy of the method are examined.Lanczos-type algorithms with embedded interpolation and extrapolation models for solving large-scale systems of linear equations.https://www.zbmath.org/1453.650602021-02-27T13:50:00+00:00"Maharani, Maharani"https://www.zbmath.org/authors/?q=ai:maharani.maharani"Larasati, Niken"https://www.zbmath.org/authors/?q=ai:larasati.niken"Salhi, Abdellah"https://www.zbmath.org/authors/?q=ai:salhi.abdellah"Khan, Wali Mashwani"https://www.zbmath.org/authors/?q=ai:khan.wali-mashwaniSummary: The new approach to combating instability in Lanczos-type algorithms for large-scale problems is proposed. It is a modification of so called the embedded interpolation and extrapolation model in Lanczos-type algorithms (EIEMLA), which enables us to interpolate the sequence of vector solutions generated by a Lanczos-type algorithm entirely, without rearranging the position of the entries of the vector solutions. The numerical results show that the new approach performs more effectively than the EIEMLA. In fact, we extend this new approach on the use of a restarting framework to obtain the convergence of Lanczos algorithms accurately. This kind of restarting challenges other existing restarting strategies in Lanczos-type algorithms.Convergence of the Crank-Nicolson method for a singularly perturbed parabolic reaction-diffusion system.https://www.zbmath.org/1453.652362021-02-27T13:50:00+00:00"Victor, Franklin"https://www.zbmath.org/authors/?q=ai:victor.franklin"Miller, John J. H."https://www.zbmath.org/authors/?q=ai:miller.john-j-h"Sigamani, Valarmathi"https://www.zbmath.org/authors/?q=ai:sigamani.valarmathiSummary: A general parabolic system of singularly perturbed linear equations of reaction-diffusion type is considered. The components of the solution exhibit overlapping layers. A numerical method with the Crank-Nicolson operator on a uniform mesh for time and classical finite difference operator on a Shishkin piecewise uniform mesh for space is constructed. It is proved that in the maximum norm, the numerical approximations obtained with this method are second order convergent in time and essentially second order convergent in space.
For the entire collection see [Zbl 1354.65008].Singularly perturbed delay differential equations and numerical methods.https://www.zbmath.org/1453.651482021-02-27T13:50:00+00:00"Narasimhan, Ramanujam"https://www.zbmath.org/authors/?q=ai:narasimhan.ramanujamSummary: The main objective of my talk is to discuss some numerical methods for singularly perturbed delay differential equations. First some well-known mathematical models represented by differential equations with out delay and with delay are presented. Then some basic numerical methods for delay differential equations are briefly described. After this an introduction to singularly perturbed delay problems is given. Finally some numerical methods for these problems are discussed.
For the entire collection see [Zbl 1354.65008].On lower bounds for optimal Jacobian accumulation.https://www.zbmath.org/1453.651102021-02-27T13:50:00+00:00"Mosenkis, Viktor"https://www.zbmath.org/authors/?q=ai:mosenkis.viktor"Naumann, Uwe"https://www.zbmath.org/authors/?q=ai:naumann.uweSummary: The task of finding a way to compute \(F'\) by using the minimal number of multiplications is generally referred to as the Optimal Jacobian Accumulation (OJA) problem. Often a special case of the \(OJA\) problem is examined, where the accumulation of the Jacobian is considered as a transformation of the linearized directed acyclic graph (l-DAG) \(G\) into \(G'\), such that \(G'\) is a subgraph of the complete directed bipartite graph \(K_{n,m}\). The transformation of the l-DAG is performed by elimination methods. The most commonly used elimination methods are vertex elimination and edge elimination. The problem of transforming an l-DAG into a bipartite graph by vertex or edge eliminations with minimal costs is generally referred as the Vertex Elimination (VE) or Edge Elimination (EE) problem. Both the VE and EE problems are conjectured to be NP-hard. Thus Branch and bound based algorithms in addition to greedy heuristics are used to solve these problems. For efficient application of such algorithms, good lower bounds are essential. In this paper, we develop a new approach for computing the lower bounds for the optimal solution of the VE and EE problems.SIMPLE adjoint message passing.https://www.zbmath.org/1453.653972021-02-27T13:50:00+00:00"Towara, M."https://www.zbmath.org/authors/?q=ai:towara.markus"Naumann, U."https://www.zbmath.org/authors/?q=ai:naumann.uweSummary: In the context of numerical simulation a system of partial differential equations is typically solved by discretizing it into a set of linear equations. The non-linear effects in the solution are then approximated by repeatedly discretizing, solving and correcting the equations using some outer iteration scheme. One example for such a scheme is the SIMPLE algorithm. In this paper, we show how we obtained an algorithmic adjoint version of the SIMPLE algorithm in OpenFOAM with particular focus on the adjoint message passing interface communication inside the linear solvers. Due to the sparse matrix storage and specifics of the implementation of the linear solvers, this method requires special attention to the inner workings of said structures and algorithms. We show the feasibility of the chosen method by presenting two case studies. In addition to the SIMPLE algorithm, the presented ideas are applicable to a wide range of algorithms whenever linear systems are embedded into an overlying non-linear context.The stability of Gauge-Uzawa method to solve nanofluid.https://www.zbmath.org/1453.652702021-02-27T13:50:00+00:00"Jang, Deok-Kyu"https://www.zbmath.org/authors/?q=ai:jang.deok-kyu"Kim, Taek-Cheol"https://www.zbmath.org/authors/?q=ai:kim.taek-cheol"Pyo, Jae-Hong"https://www.zbmath.org/authors/?q=ai:pyo.jae-hongSummary: Nanofluids is the fluids mixed with nanoscale particles and the mixed nano size materials affect heat transport. Researchers in this field has been focused on modeling and numerical computation by engineers In this paper, we analyze stability constraint of the dominant equations and check validate of the condition for most kinds of materials. So we mathematically analyze stability of the system. Also we apply Gauge-Uzawa algorithm to solve the system and prove stability of the method.Differentiating through conjugate gradient.https://www.zbmath.org/1453.650682021-02-27T13:50:00+00:00"Christianson, Bruce"https://www.zbmath.org/authors/?q=ai:christianson.bruceSummary: We show that, although the conjugate gradient (CG) algorithm has a singularity at the solution, it is possible to differentiate forward through the algorithm automatically by re-declaring all the variables as truncated Taylor series, the type of active variable widely used in automatic differentiation (AD) tools such as ADOL-C. If exact arithmetic is used, this approach gives a complete sequence of correct directional derivatives of the solution, to arbitrary order, in a single cycle of at most \(n\) iterations, where \(n\) is the number of dimensions. In the inexact case, the approach emphasizes the need for a means by which the programmer can communicate certain conditions involving derivative values directly to an AD tool.Towards a full higher order AD-based continuation and bifurcation framework.https://www.zbmath.org/1453.654392021-02-27T13:50:00+00:00"Charpentier, Isabelle"https://www.zbmath.org/authors/?q=ai:charpentier.isabelle"Cochelin, Bruno"https://www.zbmath.org/authors/?q=ai:cochelin.brunoSummary: Some of the theoretical aspects of continuation and bifurcation methods devoted to the solution for nonlinear parametric systems are presented in a higher order automatic differentiation (HOAD) framework. Besides, benefits in terms of generality and ease of use, HOAD is used to assess fold and simple bifurcations points. In particular, the formation of a geometric series in successive Taylor coefficients allows for the implementation of an efficient detection and branch switching method at simple bifurcation points. Some comparisons with the Auto and MatCont continuation software are proposed. Strengths are then exemplified on a classical case study in structural mechanics.Estimating the expansion coefficients of a geomagnetic field model using first-order derivatives of associated Legendre functions.https://www.zbmath.org/1453.860372021-02-27T13:50:00+00:00"Bücker, H. Martin"https://www.zbmath.org/authors/?q=ai:bucker.h-martin"Willkomm, Johannes"https://www.zbmath.org/authors/?q=ai:willkomm.johannesSummary: The associated Legendre functions are defined on a closed interval. Thus, their derivatives do not exist at the endpoints of the interval. However, one-sided derivatives may exist at the endpoints if the ordinary limit is replaced by a one-sided limit. When computer models that evaluate associated Legendre functions at an endpoint are transformed by automatic differentiation, an approach is needed that is tailored to one-sided derivatives. Rather than employing a black-box approach of automatic differentiation, a hierarchical approach is introduced that is based on analytic first-order one-sided derivatives of the associated Legendre functions at the endpoints. This hierarchical approach is implemented in the automatic differentiation software ADiMat and its feasibility is demonstrated in a parameter estimation problem arising from geomagnetic field modelling.Newton step methods for AD of an objective defined using implicit functions.https://www.zbmath.org/1453.651052021-02-27T13:50:00+00:00"Bell, Bradley M."https://www.zbmath.org/authors/?q=ai:bell.bradley-m"Kristensen, Kasper"https://www.zbmath.org/authors/?q=ai:kristensen.kasperSummary: We consider the problem of computing derivatives of an objective that is defined using implicit functions; i.e., implicit variables are computed by solving equations that are often nonlinear and solved by an iterative process. If one were to apply Algorithmic Differentiation (AD) directly, one would differentiate the iterative process. In this paper we present the Newton step methods for computing derivatives of the objective. These methods make it easy to take advantage of sparsity, forward mode, reverse mode, and other AD techniques. We prove that the partial Newton step method works if the number of steps is equal to the order of the derivatives. The full Newton step method obtains two derivatives order for each step except for the first step. There are alternative methods that avoid differentiating the iterative process; e.g., the method implemented in ADOL-C. An optimal control example demonstrates the advantage of the Newton step methods when computing both gradients and Hessians. We also discuss the Laplace approximation method for nonlinear mixed effects models as an example application.A benchmark of selected algorithmic differentiation tools on some problems in computer vision and machine learning.https://www.zbmath.org/1453.650502021-02-27T13:50:00+00:00"Srajer, Filip"https://www.zbmath.org/authors/?q=ai:srajer.filip"Kukelova, Zuzana"https://www.zbmath.org/authors/?q=ai:kukelova.zuzana"Fitzgibbon, Andrew"https://www.zbmath.org/authors/?q=ai:fitzgibbon.andrew-wSummary: Algorithmic differentiation (AD) allows exact computation of derivatives given only an implementation of an objective function. Although many AD tools are available, a proper and efficient implementation of AD methods is not straightforward. The existing tools are often too different to allow for a general test suite. In this paper, we compare 15 ways of computing derivatives including 11 automatic differentiation tools implementing various methods and written in various languages (C++, F\#, MATLAB, Julia and Python), 2 symbolic differentiation tools, finite differences and hand-derived computation.
We look at three objective functions from computer vision and machine learning. These objectives are for the most part simple, in the sense that no iterative loops are involved, and conditional statements are encapsulated in functions such as abs or logsumexp. However, it is important for the success of AD that such `simple' objective functions are handled efficiently, as so many problems in computer vision and machine learning are of this form.On the convergence of the accelerated Riccati iteration method.https://www.zbmath.org/1453.650852021-02-27T13:50:00+00:00"Rajasingam, Prasanthan"https://www.zbmath.org/authors/?q=ai:rajasingam.prasanthan"Xu, Jianhong"https://www.zbmath.org/authors/?q=ai:xu.jianhongSummary: In this paper, we establish results fully addressing two open problems proposed recently by \textit{I. G. Ivanov} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 11, 4012--4024 (2008; Zbl 1162.65020)], with respect to the convergence of the accelerated Riccati iteration method for solving the continuous coupled algebraic Riccati equation, or CCARE for short. These results confirm several desirable features of that method, including the monotonicity and boundedness of the sequences it produces, its capability of determining whether the CCARE has a solution, the extremal solutions it computes under certain circumstances, and its faster convergence than the regular Riccati iteration method.Using automatic differentiation for compressive sensing in uncertainty quantification.https://www.zbmath.org/1453.940322021-02-27T13:50:00+00:00"Wang, Mu"https://www.zbmath.org/authors/?q=ai:wang.mu"Lin, Guang"https://www.zbmath.org/authors/?q=ai:lin.guang"Pothen, Alex"https://www.zbmath.org/authors/?q=ai:pothen.alexSummary: This paper employs automatic differentiation (AD) in the compressive sensing-based generalized polynomial chaos (gPC) expansion, which computes a sparse approximation of the Quantity of Interest (QoI) using orthogonal polynomials as basis functions. An earlier approach without AD relies on an iterative procedure to refine the solution by approximating the gradient of the QoI. With AD, the gradient can be accurately evaluated, and a set of basis functions of the gPC expansion associated with new random variables can be efficiently identified. The computational complexity of the algorithm using AD is independent of the number of basis functions, whereas an earlier algorithm had complexity proportional to the square of this number. Our test problems include synthetic problems and a high-dimensional stochastic partial differential equation. With the new basis, the coefficient vector in the gPC expansion is sparser than the original basis. We demonstrate that introducing AD can greatly improve the performance by computing solutions 2 to 10 times faster than an earlier approach. The accuracy of the gPC expansion is also improved; sparse gpC expansions are obtained without iterative refinement, even for high dimensions when an earlier approach fails.Implementation of second derivative general linear methods.https://www.zbmath.org/1453.651552021-02-27T13:50:00+00:00"Abdi, Ali"https://www.zbmath.org/authors/?q=ai:abdi.ali|abdi.ali-z"Conte, Dajana"https://www.zbmath.org/authors/?q=ai:conte.dajanaSummary: In this paper, the implementation of second derivative general linear methods (SGLMs) in a variable stepsize environment using Nordsieck technique is discussed and various implementation issues are investigated. All coefficients of a method of order four together with its error estimate are obtained. The method is derived with the aim of good zero-stability properties for a large range of ratios of sequential stepsizes to implement in a variable stepsize mode. The numerical experiments indicate that the constructed error estimate is very reliable in a variable stepsize environment and beautifully confirm the efficiency and robustness of the proposed scheme based on SGLMs. Moreover, the results verify that the proposed scheme outperforms the code \textsf{ode15s} from \textrm{Matlab} ODE suite on systems whose Jacobian has eigenvalues which are close to the imaginary axis.Stochastic control for optimal execution: fast approximation solution scheme under nested mean-semi deviation and conditional value at risk.https://www.zbmath.org/1453.911092021-02-27T13:50:00+00:00"He, Meng-Fei"https://www.zbmath.org/authors/?q=ai:he.meng-fei"Li, Duan"https://www.zbmath.org/authors/?q=ai:li.duan"Chen, Yuan-Yuan"https://www.zbmath.org/authors/?q=ai:chen.yuanyuan|chen.yuanyuan.1This paper considers the optimal execution problem of completing a large volume trading order within a given trading horizon with a goal to minimize the price market. The mathematical model is formulated and the risk measures is proposed in the study. An approximation scheme is developed to make the solution process tractable. The performance of the proposed strategy is demonstrated by some numerical experiments.
Reviewer: Hang Lau (Montréal)Convergence rates of damped inertial dynamics under geometric conditions and perturbations.https://www.zbmath.org/1453.340762021-02-27T13:50:00+00:00"Sebbouh, O."https://www.zbmath.org/authors/?q=ai:sebbouh.o"Dossal, Ch."https://www.zbmath.org/authors/?q=ai:dossal.charles"Rondepierre, A."https://www.zbmath.org/authors/?q=ai:rondepierre.audeThis paper investigates a family of second-order ODEs associated with the inertial gradient descent. It appears that such ODEs are widely used to build trajectories converging to a minimizer \(x^\ast\) of a (possibly convex) function \(F\). Furthermore, the considered family includes the continuous version of a Nesterov inertial scheme and the continuous heavy ball method. Thus, several damping parameters, which are not necessarily vanishing, as well as a perturbation term \(g\) are considered. Here, the damping parameter is linked to the inertia of the associated inertial scheme, while the perturbation term \(g\) is linked to the error that can be made on the gradient of the function \(F\). The authors present new asymptotic bounds on \(F(x(t))-F(x^\ast)\), where \(x\) denotes a solution of the ODE, provided \(F\) is convex and satisfies local geometrical properties such as of Łojasiewicz-type and under integrability conditions on \(g\). Although geometrical properties and perturbations were already investigated for most ODEs of these families, they are now jointly studied. Such results yield an insight on the behavior of these inertial and perturbed algorithms in case \(F\) satisfies some Łojasiewicz properties especially in the setting of stochastic algorithms.
Reviewer: Christian Pötzsche (Klagenfurt)Localized method of fundamental solutions for 2D harmonic elastic wave problems.https://www.zbmath.org/1453.740432021-02-27T13:50:00+00:00"Li, Weiwei"https://www.zbmath.org/authors/?q=ai:li.weiweiSummary: In this paper, a localized version of the method of fundamental solutions (MFS) named as the localized MFS (LMFS) is applied to two-dimensional (2D) harmonic elastic wave problems. Due to the dense and ill-conditioned coefficient matrices, the traditional MFS is computationally expensive and time-consuming to solve the above-mentioned problems which require a large amount number of nodes to obtain excellent numerical results. In the LMFS, the domain is divided into small subdomains, in which the traditional MFS is applied to represent the physical variables as linear combinations of the fundamental solution. A sparse and banded system of linear equations is yielded in this method, so that it is attractive to solve large-scale problems. A numerical example demonstrates the effectiveness and accuracy of the proposed LMFS.A regularized method of moments for three-dimensional time-harmonic electromagnetic scattering.https://www.zbmath.org/1453.780042021-02-27T13:50:00+00:00"Li, Junpu"https://www.zbmath.org/authors/?q=ai:li.junpu"Zhang, Lan"https://www.zbmath.org/authors/?q=ai:zhang.lan"Qin, Qing-Hua"https://www.zbmath.org/authors/?q=ai:qin.qinghuaSummary: A regularized method of moments based on the modified fundamental solution of the Helmholtz equation is proposed in this article. The regularized method of moments uses the origin intensity factor technique which is free of mesh and integration to deal with the singularity at origin of the basis function. Thus, the time-consuming singular integration can be avoided. In addition, the non-uniqueness at internal resonance is also fixed using the constructed modified fundamental solution. In comparison with the traditional method of moments, the regularized method of moments can reduce the computational time by half, while the stability and accuracy stay about the same. Numerical experiments demonstrate that the regularized method of moments can accurately and efficiently compute the radar cross section of perfect conducting scatter in all frequency ranges.An asymptotic approximation for the permanent of a doubly stochastic matrix.https://www.zbmath.org/1453.650832021-02-27T13:50:00+00:00"McCullagh, Peter"https://www.zbmath.org/authors/?q=ai:mccullagh.peterSummary: A determinantal approximation is obtained for the permanent of a doubly stochastic matrix. For moderate-deviation matrix sequences, the asymptotic relative error is of order \(O(n^{-1})\).Superdiffusion in the presence of a reflecting boundary.https://www.zbmath.org/1453.351772021-02-27T13:50:00+00:00"Jesus, Carla"https://www.zbmath.org/authors/?q=ai:jesus.carla"Sousa, Ercília"https://www.zbmath.org/authors/?q=ai:sousa.erciliaSummary: We study the effect of having a reflecting boundary condition in a superdiffusive model. Firstly it is described how the problem formulation is affected by this type of physical boundary and then it is shown how to implement an implicit numerical method to compute the numerical solutions. The consistency and stability analysis of the numerical method are discussed. In the end numerical experiments are presented to show the performance of the scheme and to visualize the consequences of having a reflecting wall.The study of exact and numerical solutions of the generalized viscous Burgers' equation.https://www.zbmath.org/1453.652402021-02-27T13:50:00+00:00"Zhang, Qifeng"https://www.zbmath.org/authors/?q=ai:zhang.qifeng|zhang.qifeng.1"Qin, Yifan"https://www.zbmath.org/authors/?q=ai:qin.yifan"Wang, Xuping"https://www.zbmath.org/authors/?q=ai:wang.xuping"Sun, Zhi-zhong"https://www.zbmath.org/authors/?q=ai:sun.zhizhongSummary: In the paper, we revisit the uniform boundedness of the exact solution and then study pointwise error estimate for two kinds of conservative numerical discretization schemes. We show that both difference schemes share the conservative invariants similar to the original continuous model for all positive integers \(p\). In particular, we prove that difference schemes are convergent in pointwise sense based on the cut-off function method. A numerical example is carried out to verify our theoretical results.New finite difference pair with optimized phase and stability properties.https://www.zbmath.org/1453.651622021-02-27T13:50:00+00:00"Yao, Junfeng"https://www.zbmath.org/authors/?q=ai:yao.junfeng"Simos, T. E."https://www.zbmath.org/authors/?q=ai:simos.theodore-eSummary: In this paper and for the first time in the literature, a new four-stages symmetric two-step finite difference pair with optimized phase and stability properties is introduced. The new scheme has the following characteristics:
\begin{itemize}
\item[1.] is of symmetric type,
\item[2.] is of two-step algorithm,
\item[3.] is of four-stages,
\item[4.] is of tenth-algebraic order,
\item[5.] the approximations which produces the new finite difference pair are the following:
\begin{itemize}
\item[--] An approximation developed on the first layer on the point \(x_{n-1}\),
\item[--] An approximation developed on the second layer on the point \(x_{n-1}\),
\item[--] An approximation developed on the third layer on the point \(x_{n}\) and finally,
\item[--] An approximation developed on the fourth (final) layer on the point \(x_{n+1}\),
\end{itemize}
\item[6.] it has eliminated the phase-lag and its first, second, third and fourth derivatives,
\item[7.] it has optimized stability properties,
\item[8.] is a P-stable methods since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).
\end{itemize}
For the new finite difference scheme we describe an error and stability analysis. The examination of the efficiency of the new obtained finite difference pair is based on its application on systems of coupled differential equations arising from the Schrödinger equation.A hybrid finite difference pair with maximum phase and stability properties.https://www.zbmath.org/1453.651582021-02-27T13:50:00+00:00"Fang, Jie"https://www.zbmath.org/authors/?q=ai:fang.jie"Liu, Chenglian"https://www.zbmath.org/authors/?q=ai:liu.chenglian"Simos, T. E."https://www.zbmath.org/authors/?q=ai:simos.theodore-eSummary: We develop, in the present paper and for the first time in the literature, a new hybrid finite difference pair of symmetric two-step. The basic properties of the new pair are:
\begin{itemize}
\item[--] is of symmetric form,
\item[--] is of two-step,
\item[--] is of four-stages,
\item[--] is of tenth-algebraic order,
\item[--] the development of the hybrid symmetric two-step pair is of the following form:
\begin{itemize}
\item[{\(\bullet\)}] first layer is an approximation on the point \(x_{n-1}\),
\item[{\(\bullet\)}] second layer is an approximation on the point \(x_{n-1}\),
\item[{\(\bullet\)}] third layer is an approximation on the point \(x_{n}\) and finally,
\item[{\(\bullet\)}] fourth layer is an approximation on the point \(x_{n+1}\),
\end{itemize}
\item[--] it has vanished the phase-lag and its first, second and third derivatives,
\item[--] it has excellent stability properties,
\item[--] it has an interval of periodicity equal to \(\left( 0, \infty \right) \).
\end{itemize}
For the new symmetric finite difference pair a full analysis is presented. We evaluate the efficiency of the new obtained symmetric finite difference pair by applying it on systems of coupled differential equations of the Schrödinger form.Fast exponential time differencing/spectral-Galerkin method for the nonlinear fractional Ginzburg-Landau equation with fractional Laplacian in unbounded domain.https://www.zbmath.org/1453.653652021-02-27T13:50:00+00:00"Wang, Pengde"https://www.zbmath.org/authors/?q=ai:wang.pengdeSummary: This paper proposes a fast and efficient spectral-Galerkin method for the nonlinear complex Ginzburg-Landau equation involving the fractional Laplacian in \(\mathbb{R}^d\). By employing the Fourier-like bi-orthogonal mapped Chebyshev function as basis functions, the fractional Laplacian can be fully diagonalized. Then for the resulting diagonalized semi-discrete system, an exponential time differencing scheme is proposed for the temporal discretization. The obtained method can be fast implemented and has second order accuracy in time and algebraical accuracy in space. One- and two-dimensional numerical examples are tested to validate the accuracy and efficiency of the proposed method.A hybrid cooperative cuckoo search algorithm with particle swarm optimisation.https://www.zbmath.org/1453.902162021-02-27T13:50:00+00:00"Wang, Lijin"https://www.zbmath.org/authors/?q=ai:wang.lijin"Zhong, Yiwen"https://www.zbmath.org/authors/?q=ai:zhong.yiwen"Yin, Yilong"https://www.zbmath.org/authors/?q=ai:yin.yilongSummary: This paper proposes an improved hybrid cooperative algorithm that combines cooperative cuckoo search algorithm and particle swarm optimisation, called HCCSPSO. The cooperative co-evolutionary framework is applied to cuckoo search algorithm to implement dimensional cooperation. The particle swarm optimisation algorithm, viewed as a cooperative component, is embedded in the back of the cuckoo search algorithm. During iteration, the best solution obtained by the previous cooperative component is randomly embedded in the last one to avoid the pseudo-minima produced by the previous one, while the subcomponents of best solution from the last cooperative component are also randomly planted in the subcomponents of the previous one. The results of experimental simulations demonstrate the improvement in the efficiency and the effect of the cooperation strategy, and the promising of HCCSPSO.Simulation reductions for the Ising model.https://www.zbmath.org/1453.820092021-02-27T13:50:00+00:00"Huber, Mark"https://www.zbmath.org/authors/?q=ai:huber.mark-lSummary: Polynomial time reductions between problems have long been used to delineate problem classes, where an oracle for solving one problem yields a solution to another. Simulation reductions also exist, where an oracle for simulation from a probability distribution is employed in order to obtain draws from another distribution. Here linear time simulation reductions are given for: the Ising spins world to the Ising subgraphs world and the Ising subgraphs world to the Ising spins world. This answers a long standing question of whether such a direct relationship between these two versions of the Ising model existed. Moreover, these reductions result in the first method for perfect simulation from the subgraphs world and a new Swendsen-Wang style Markov chain for the Ising model. The method used is to write the desired distribution with set parameters as a mixture of distributions where the parameters are at their extreme values.New error analysis of a second order BDF scheme for unsteady natural convection problem.https://www.zbmath.org/1453.760792021-02-27T13:50:00+00:00"Liu, Qian"https://www.zbmath.org/authors/?q=ai:liu.qian"Shi, Dongyang"https://www.zbmath.org/authors/?q=ai:shi.dongyangSummary: In this paper, superconvergence analysis of a mixed finite element method (MFEM) is proposed for the unsteady natural convection problem. The system is discretized by the \(Q_{11} / Q_0\) conforming finite elements in space and the linearized second-order BDF method in time. Based on the high accuracy characters of the elements, the superclose and global superconvergence results of order \(\mathcal{O}( h^2 + \tau^2)\) for the velocity \(\mathbf{u}\), the temperature \(T\) in \(H^1\)-norm and the pressure \(p\) in \(L^2\)-norm are deduced rigorously. Here, \(h\) is the subdivision parameter and \(\tau\), the time step. The main difficulties caused by the complicate convection terms and the couplings between \(\mathbf{u}, p\) and \(T\) are treated with new skillful techniques. Numerical results are provided to confirm the theoretical analysis.Sinc-Galerkin method for solving higher order fractional boundary value problems.https://www.zbmath.org/1453.651912021-02-27T13:50:00+00:00"Darweesh, A."https://www.zbmath.org/authors/?q=ai:darweesh.amer-hSummary: In this work we use the sinc-Galerkin method to solve higher order fractional boundary value problems. We estimate the second order fractional derivative in the Caputo sense. More precisely, we find a numerical solution for
\[
\begin{gathered}
g_1(t)D^\alpha u(t)+g_2(t)D^\beta u(t)+ p(t)u^{(4)}(t)+q(t)u(t)=f(t),\\
0<t<1,\quad 0<\beta<1,\quad 1<\alpha<2,
\end{gathered}
\]
subject to the boundary conditions \(u(0)=0\), \(u'(0)=0\), \(u(1)=0\), \(u'(1)=0\). Our contribution appears in the estimate of \(D^\alpha u\) for higher order \(\alpha\). Numerical examples are described to show the accuracy of this attempt where we applied the sinc-Galerkin method for fractional order differential equations with singularities.A survey on true random number generators based on chaos.https://www.zbmath.org/1453.650142021-02-27T13:50:00+00:00"Yu, Fei"https://www.zbmath.org/authors/?q=ai:yu.fei"Li, Lixiang"https://www.zbmath.org/authors/?q=ai:li.lixiang"Tang, Qiang"https://www.zbmath.org/authors/?q=ai:tang.qiang"Cai, Shuo"https://www.zbmath.org/authors/?q=ai:cai.shuo"Song, Yun"https://www.zbmath.org/authors/?q=ai:song.yun"Xu, Quan"https://www.zbmath.org/authors/?q=ai:xu.quanSummary: With the rapid development of communication technology and the popularization of network, information security has been highly valued by all walks of life. Random numbers are used in many cryptographic protocols, key management, identity authentication, image encryption, and so on. True random numbers (TRNs) have better randomness and unpredictability in encryption and key than pseudorandom numbers (PRNs). Chaos has good features of sensitive dependence on initial conditions, randomness, periodicity, and reproduction. These demands coincide with the rise of TRNs generating approaches in chaos field. This survey paper intends to provide a systematic review of true random number generators (TRNGs) based on chaos. Firstly, the two kinds of popular chaotic systems for generating TRNs based on chaos, including continuous time chaotic system and discrete time chaotic system are introduced. The main approaches and challenges are exposed to help researchers decide which are the ones that best suit their needs and goals. Then, existing methods are reviewed, highlighting their contributions and their significance in the field. We also devote a part of the paper to review TRNGs based on current-mode chaos for this problem. Finally, quantitative results are given for the described methods in which they were evaluated, following up with a discussion of the results. At last, we point out a set of promising future works and draw our own conclusions about the state of the art of TRNGs based on chaos.Optimal convergence rates for Nesterov acceleration.https://www.zbmath.org/1453.901172021-02-27T13:50:00+00:00"Aujol, Jean-Francois"https://www.zbmath.org/authors/?q=ai:aujol.jean-francois"Dossal, Charles"https://www.zbmath.org/authors/?q=ai:dossal.charles"Rondepierre, Aude"https://www.zbmath.org/authors/?q=ai:rondepierre.audeUsing kernel-based collocation methods to solve a delay partial differential equation with application to finance.https://www.zbmath.org/1453.653542021-02-27T13:50:00+00:00"Azari, Hossein"https://www.zbmath.org/authors/?q=ai:azari.hossein"Moradipour, Mojtaba"https://www.zbmath.org/authors/?q=ai:moradipour.mojtabaSummary: We consider a delay partial differential equation arising in a jump diffusion model of option pricing. Under the mean-reverting jump-diffusion model, the price of options on electricity satisfies a second order partial differential equation. In this paper, we use positive definite kernels to discretise the PDE in spatial direction and achieve a linear system of first order differential equation with respect to time. We impose homogenous boundary conditions of the PDE by using a manipulated version of kernels called `recursive kernels'. The proposed methods are fast and accurate with low computational complexity. No integrations are necessary and the time dependent system of differential equations can be solving analytically. Illustrative example is included to demonstrate the validity and applicability of the new techniques.Spectrum of Dirichlet BDIDE operator.https://www.zbmath.org/1453.653982021-02-27T13:50:00+00:00"Mohamed, N. A."https://www.zbmath.org/authors/?q=ai:mohamed.nurul-akmal"Ibrahim, N. F."https://www.zbmath.org/authors/?q=ai:ibrahim.nur-fadhilah"Mohamed, N. F."https://www.zbmath.org/authors/?q=ai:faried.nashat"Mohamed, N. H."https://www.zbmath.org/authors/?q=ai:mohamed.nurul-hudaSummary: In this paper, we present the distribution of some maximal eigenvalues that are obtained numerically from the discrete Dirichlet Boundary Domain Integro-Dierential Equation (BDIDE) operator. We also discuss the convergence of the discrete Dirichlet BDIDE that corresponds with the obtained absolute value of the largest eigenvalues of the discrete BDIDE operator. There are three test domains that are considered in this paper, i.e., a square, a circle, and a parallelogram. In our numerical test, the eigenvalues disperse as the power of the variable coecient increases. Not only that, we also note that the dispersion of the eigenvalues corresponds with the characteristic size of the test domains. It enables us to predict the convergence of an iterative method. This is an advantage as it enables the use of an iterative method in solving Dirichlet BDIDE as an alternative to the direct methods.Mean field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods.https://www.zbmath.org/1453.351652021-02-27T13:50:00+00:00"Gomes, S. N."https://www.zbmath.org/authors/?q=ai:gomes.susana-n"Pavliotis, G. A."https://www.zbmath.org/authors/?q=ai:pavliotis.grigorios-a"Vaes, U."https://www.zbmath.org/authors/?q=ai:vaes.uThe performance of higher moments estimators: an empirical study.https://www.zbmath.org/1453.625112021-02-27T13:50:00+00:00"Harun, H. F."https://www.zbmath.org/authors/?q=ai:harun.h-f"Abdullah, M. H."https://www.zbmath.org/authors/?q=ai:abdullah.m-kh|abdullah.mimi-hafizahSummary: This study investigates the performance of higher order moments, realised from the model-free Bakshi-Kapadia-Madan (MFBKM). We concentrate on investigating higher order option-implied moments â variance, skewness and kurtosis, chosen in relation to contracts dened in MFBKM, i.e. volatility, cubic, and quartic contract. The three approaches adopted in order to estimate the integrals of the dened MFBKM contracts are the basic (trapezoidal-rule), adapted (single-combined) and advanced method (cubic-spline). The sample data is extracted from DJIA index options data, which covers the period from January 2009 until December 2015. The results show that the advanced method performs poorly in estimating the MFBKM, especially in the case of skewness and kurtosis integrals estimation. The advanced method outperforms the other approaches in the case of the variance estimation. In estimating both model-free skewness and kurtosis, the adapted method is found to perform the best, instead.A time multidomain spectral method for valuing affine stochastic volatility and jump diffusion models.https://www.zbmath.org/1453.653612021-02-27T13:50:00+00:00"Moutsinga, Claude Rodrigue Bambe"https://www.zbmath.org/authors/?q=ai:moutsinga.claude-rodrigue-bambe"Pindza, Edson"https://www.zbmath.org/authors/?q=ai:pindza.edson"Maré, Eben"https://www.zbmath.org/authors/?q=ai:mare.ebenA time-spectral domain decomposition method is developed that accommodates differential equations arising from financial models of affine type. The affine structure of the financial models is used to avoid solving the multi-dimensional partial integro-differential equation (PIDE) but rather to solve a system of Riccati equations. The method is based on the Tau-matrix approach using a differentiation matrix method on a time interval divided into disjoint domains. The Riccati equations are solved in the frequency domain using an operational matrix based on Chebyshev polynomials. In this way, the original problem is transformed into an iterative system of algebraic equations that is easier to solve. Three numerical examples are implemented and solutions are compared to numerical solutions from Chebfun [\textit{R. B. Platte} and \textit{L. N. Trefethen}, Math. Ind. 15, 69--87 (2010; Zbl 1220.65100)]. The numerical results show that the method maintains its spectral convergence even for large time-space intervals. The method can be applied to other affine models with jumps.
Reviewer: Bülent Karasözen (Ankara)A note on generalized averaged Gaussian formulas for a class of weight functions.https://www.zbmath.org/1453.650562021-02-27T13:50:00+00:00"Spalević, Miodrag M."https://www.zbmath.org/authors/?q=ai:spalevic.miodrag-mSummary: In the recent paper [\textit{S. E. Notaris}, Numer. Math. 142, No. 1, 129--147 (2019; Zbl 1411.41022)] it has been introduced a new and useful class of nonnegative measures for which the well-known Gauss-Kronrod quadrature formulae coincide with the generalized averaged Gaussian quadrature formulas. In such a case, the given generalized averaged Gaussian quadrature formulas are of the higher degree of precision, and can be numerically constructed by an effective and simple method; see [\textit{M. M. Spalević}, Math. Comput. 76, No. 259, 1483--1492 (2007; Zbl 1113.65025)]. Moreover, as almost immediate consequence of our results from [Spalević, loc. cit.] and that theory, we prove the main statements in [Notaris, loc. cit.] in a different manner, by means of the Jacobi tridiagonal matrix approach.A splitting mixed covolume method for viscoelastic wave equations on triangular grids.https://www.zbmath.org/1453.652632021-02-27T13:50:00+00:00"Zhao, Jie"https://www.zbmath.org/authors/?q=ai:zhao.jie"Li, Hong"https://www.zbmath.org/authors/?q=ai:li.hong"Fang, Zhichao"https://www.zbmath.org/authors/?q=ai:fang.zhichao"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.11"Wang, Huifang"https://www.zbmath.org/authors/?q=ai:wang.huifangThe authors propose a new splitting mixed covolume (SMCV) method for solving viscoelastic wave equations. The method combines the splitting positive definite mixed finite element (SPDMFE) and the mixed covolume (MCV) methods. One of the main features of this new scheme is that it does not require to solve the coupled system, thus reducing the scale of linear equations. This stems from the idea of SPDMFE. The semi-discrete and fully discrete SMCV schemes are established thanks to the introduction of a transfer operator. The existence and uniqueness for the semi-discrete scheme are justified. For both semi-discrete and fully discrete schemes, optimal a priori error estimates in different norms are proved. Some numerical tests are presented to show the efficiency of the method.
Reviewer: Abdallah Bradji (Annaba)Method of fictitious domains and homotopy as a new alternative to multidimensional partial differential equations in domains of any shape.https://www.zbmath.org/1453.654342021-02-27T13:50:00+00:00"Gavrilyuk, I. P."https://www.zbmath.org/authors/?q=ai:gavrilyuk.ivan-p"Makarov, V. L."https://www.zbmath.org/authors/?q=ai:makarov.volodymyr-lThe authors investigate a newly developed coupled method of fictitious domains and homotopy for solving multidimensional partial differential equations in domains of any shape. The complexity of dealing with the geometries of these domains has caused a real challenge associated with solving such equations; therefore, there is a great need for an alternative efficient method to overcome this challenge. On the one hand, the authors first apply the method of fictitious domains to the studied model problem, where this problem can be approximated by another problem in a canonical domain known as parallelepiped. On the other hand, the homotopy method is applied, so the studied problem in canonical domain is transferred to a more complicated form of this problem in a domain of any shape using an artificially proposed parameter with subsequent algorithm application as mentioned by the authors. Various cases are discussed using the following boundary value problem (model problem): Given \(\Omega\) as an arbitrary domain, then we have the following model problem:
\[
Lu(x) \equiv -\sum_{i,j=1}^{n} \frac{\partial}{\partial x_{i}} \left(a_{ij}(x) \frac{\partial u}{\partial x_{j}} \right)+c(x)u = f(x),
\]
where \(x=(x_{1},\dots,x_{n})\in \Omega\). This problem is subjected to the following conditions: \(u(x)=0\), where \(x \in \partial \Omega\); \(a_{ij}(x)=a_{ji}(x)\), where \(c(x) \geq 0\); \(\inf_{x \in \Omega} \sum_{i,j=1}^{n} a_{ij}(x)\xi_{i}\xi_{j} \geq \mu \sum_{i=1}^{n} \xi^{2}_{i}\), where \(\mu\) is a positive constant which is an independent of an arbitrary vector \(\xi=(\xi_{1},\dots,\xi_{n})\). The authors obtain an exponentially convergent sequence of solutions to the solution of the studied boundary value problem in parallelepiped. Finally, the proposed method is discussed in a very simple way and also applied to various problems and cases such as spectral problems, and polygonal domains in 2D and in 1D.
Reviewer: Mohammed Kaabar (Gelugor)Uncollided flux techniques for arbitrary finite element meshes.https://www.zbmath.org/1453.654072021-02-27T13:50:00+00:00"Hanuš, Milan"https://www.zbmath.org/authors/?q=ai:hanus.milan"Harbour, Logan H."https://www.zbmath.org/authors/?q=ai:harbour.logan-h"Ragusa, Jean C."https://www.zbmath.org/authors/?q=ai:ragusa.jean-c"Adams, Michael P."https://www.zbmath.org/authors/?q=ai:adams.michael-p"Adams, Marvin L."https://www.zbmath.org/authors/?q=ai:adams.marvin-lSummary: The uncollided angular flux can be difficult to compute accurately in discrete-ordinate radiation transport codes, especially in weakly-scattering configurations with localized sources. It has long been recognized that an analytical or semi-analytical treatment of the uncollided flux, coupled with a discrete-ordinate solution for the collided flux, can yield dramatic improvements in solution accuracy and computational efficiency. In this paper, we present such an algorithm for the semi-analytical calculation of the uncollided flux. This algorithm is unique in several aspects: (1) it applies to arbitrary polyhedral cells (and can be thus coupled with collided flux solvers that support arbitrary polyhedral meshes without the need for explicit tetrahedral re-meshing), (2) it provides accurate uncollided solutions near sources, (3) it is devised with parallel implementation in mind, and (4) it minimizes the total number of traced rays and maintains a reasonable ray density on each local subdomain. This paper provides a complete derivation of the algorithm and demonstrates its important features on a set of simple examples and a standard transport benchmark. Assessment of its parallel performance will be the subject of a subsequent paper.Neural network solution of pantograph type differential equations.https://www.zbmath.org/1453.651652021-02-27T13:50:00+00:00"Hou, Chih-Chun"https://www.zbmath.org/authors/?q=ai:hou.chih-chun"Simos, Theodore E."https://www.zbmath.org/authors/?q=ai:simos.theodore-e"Famelis, Ioannis Th."https://www.zbmath.org/authors/?q=ai:famelis.ioannis-thSummary: We investigate the approximate solution of pantograph type functional differential equations using neural networks. The methodology is based on the ideas of Lagaris et al., and it is applied to various problems with a proportional delay term subject to initial or boundary conditions. The proposed methodology proves to be very efficient.Algebraic solution of fuzzy linear system as: \(\widetilde{A}\widetilde{X}+\widetilde{B}\widetilde{X}=\widetilde{Y}\).https://www.zbmath.org/1453.650942021-02-27T13:50:00+00:00"Allahviranloo, T."https://www.zbmath.org/authors/?q=ai:allahviranloo.tofigh"Babakordi, F."https://www.zbmath.org/authors/?q=ai:babakordi.fThe fuzzy system \(\widetilde{A}\widetilde{X}+\widetilde{B}\widetilde{X}=\widetilde{Y}\) has square matrices \(\widetilde{A}\) and \(\widetilde{B}\) and vectors \(\widetilde{X}\) and \(\widetilde{B}\). If all their entries are intervals, then this is an interval linear system (ILS) with matrices and vectors denoted with square brackets \([\,\cdot\,]\). For example \([X]\) has entries \([x_i]=[\underline{x}_i,\overline{x}_i]\). This ILS can be solved using an inclusion ILS (IILS), that has a solution set \(SS\), such that for all \(X\in SS\) it holds that \([A]X+[B]X\in[Y]\). By exploring the boundaries in each coordinate direction of the IILS, a solution for the ILS can be obtained. A more general fuzzy system is then solved replacing the fuzzy variables by intervals depending on a parameter \(r\in[0,1]\) like \(\widetilde{u}\in [u]_r=[m-\alpha L(r),m+\beta R(r)]\), \(\alpha,\beta>0\) and where \(L\) and \(R\) are appropriate smoothing functions decreasing monotonically from 1 to 0. Proofs are give and many numerical examples are used to illustrate the ideas. However English as well as the mathematics could have been edited much better.
Reviewer: Adhemar Bultheel (Leuven)Simulated annealing for the bounds of Kendall's \(\tau\) and Spearman's \(\rho \).https://www.zbmath.org/1453.625302021-02-27T13:50:00+00:00"Shao, Wei"https://www.zbmath.org/authors/?q=ai:shao.wei"Guo, Guangbao"https://www.zbmath.org/authors/?q=ai:guo.guangbao"Zhao, Guoqing"https://www.zbmath.org/authors/?q=ai:zhao.guoqing"Meng, Fanyu"https://www.zbmath.org/authors/?q=ai:meng.fanyuSummary: It has long been known that, for many joint distributions, Kendall's \(\tau\) and Spearman's \(\rho\) have different values, as they measure different aspects of the dependence structure. Although the classical inequalities between Kendall's \(\tau\) and Spearman's \(\rho\) for pairs of random variables are given, the joint distributions which can attain the bounds between Kendall's \(\tau\) and Spearman's \(\rho\) are difficult to find. We use the simulated annealing method to find the bounds for \(\rho\) in terms of \(\tau\) and its corresponding joint distribution which can attain those bounds. Furthermore, using this same method, we find the improved bounds between \(\tau\) and \(\rho\), which is different from that given by Durbin and Stuart.Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems.https://www.zbmath.org/1453.651692021-02-27T13:50:00+00:00"Arqub, Omar Abu"https://www.zbmath.org/authors/?q=ai:arqub.omar-abu"Al-Smadi, Mohammed"https://www.zbmath.org/authors/?q=ai:al-smadi.mohammed-h"Momani, Shaher"https://www.zbmath.org/authors/?q=ai:momani.shaher-m"Hayat, Tasawar"https://www.zbmath.org/authors/?q=ai:hayat.tasawarThe paper is concerned with the analytic and numerical solution of the two-point fuzzy boundary value problem
\[
y''(x) + f \left(x, y(x), y'(x) \right)= g \left(x, y(x), y'(x) \right) \text{ on } (a,b)
\]
subject to boundary conditions
\[
y(a)=\alpha, \quad y(b)=\beta,
\]
where \(a\) and \(b\) are real numbers, \(\alpha\) and \(\beta\) are fuzzy numbers, and \(f,g,y\) are fuzzy-valued functions. First, the concept of fuzzy analysis is briefly reviewed. Then the problem is reformulated using \(r\)-cut representations of functions under the assumption of strongly generalized differentiability and appropriate Hilbert spaces with reproducing kernels are introduced. An orthonormal system of functions is constructed and it is used for the representation of the analytic solution as an infinite sum of functions and an approximate solution as a finite sum of functions. These solutions can be computed directly in linear cases. For nonlinear equations the authors propose an iterative method and prove its convergence. Numerical experiments and a numerical comparison with the results computed by the homotopy analysis method and the Adomian decomposition method are presented.
Reviewer: Dana Černá (Liberec)Convergence rates of full-implicit truncated Euler-Maruyama method for stochastic differential equations.https://www.zbmath.org/1453.650252021-02-27T13:50:00+00:00"Wen, Haining"https://www.zbmath.org/authors/?q=ai:wen.hainingSummary: Motivated by truncated Euler-Maruyama (EM) method established by \textit{X. Mao} [J. Comput. Appl. Math. 290, 370--384 (2015; Zbl 1330.65016); ibid. 296, 362--375 (2016; Zbl 1378.65036)], a state-of-the-art scheme named full-implicit truncated EM method is derived in this paper, aiming to use shorter runtime and larger stepsize, as well as to obtain better stability property. Weaker restrictions on truncated functions of full-implicit truncated EM scheme have been obtained, which solve the disadvantage of Mao [2015, loc. cit.] requiring the stepsize to be so small that sometimes the truncated EM would be inapplicable. The superiority of our results will be highlighted by the comparisons with the achievements in [Mao 2015, 2016, loc. cit.] as well as others on the implicit Euler scheme and semi-implicit truncated EM scheme. Numerical examples verify the order of \(L^q\)-convergence, cheaper computational costs, wider stepsize and better stability.Computation of an infinite integral using integration by parts.https://www.zbmath.org/1453.260042021-02-27T13:50:00+00:00"Tang, Jian-guo"https://www.zbmath.org/authors/?q=ai:tang.jianguoSummary: In this paper, an infinite integral concerning numerical computation in crystallography is investigated, which was studied in two recent articles, and integration by parts is employed for calculating this typical integral. A variable transformation and a single integration by parts lead to a new formula for this integral, and at this time, it becomes a completely definite integral. Using integration by parts iteratively, the singularity at the points near three points \(a = 0,1,2\) can be eliminated in terms containing obtained integrals, and the factors of amplifying round-off error are released into two simple fractions independent of the integral. Series expansions for this integral are obtained, and estimations of its remainders are given, which show that accuracy \(2^{-n}\) is achieved in about \(2n\) operations for every value in a given domain. Finally, numerical results are given to verify error analysis, which coincide well with the theoretical results.The theta-Galerkin finite element method for coupled systems resulting from microsensor thermistor problems.https://www.zbmath.org/1453.653392021-02-27T13:50:00+00:00"Mbehou, Mohamed"https://www.zbmath.org/authors/?q=ai:mbehou.mohamedSummary: This paper is devoted to the analysis of a linearized theta-Galerkin finite element method for the time-dependent coupled systems resulting from microsensor thermistor problems. Hereby, we focus on time discretization based on \(\theta\)-time stepping scheme with \(\theta \in [\frac {1} {2}, 1)\) including the standard Crank-Nicolson \((\theta = \frac {1}{2})\) and the shifted Crank - Nicolson \((\theta = \frac {1}{2} + \delta,\) where \(\delta\) is the time-step) schemes. The semidiscrete formulation in space is presented and optimal error bounds in \(L^2\)-norm and the energy norm are established. For the fully discrete system, the optimal error estimates are derived for the standard Crank-Nicolson, the shifted Crank-Nicolson, and the general case where \(\theta \neq \frac {1}{2} + k \delta\) with \(k = 0,1\). Finally, numerical simulations that validate the theoretical findings are exhibited.Damped Newton's method on Riemannian manifolds.https://www.zbmath.org/1453.901612021-02-27T13:50:00+00:00"Bortoloti, Marcio Antônio de A."https://www.zbmath.org/authors/?q=ai:bortoloti.marcio-antonio-de-a"Fernandes, Teles A."https://www.zbmath.org/authors/?q=ai:fernandes.teles-a"Ferreira, Orizon P."https://www.zbmath.org/authors/?q=ai:ferreira.orizon-pereira"Yuan, Jinyun"https://www.zbmath.org/authors/?q=ai:yuan.jinyunSummary: A damped Newton's method to find a singularity of a vector field in Riemannian setting is presented with global convergence study. It is ensured that the sequence generated by the proposed method reduces to a sequence generated by the Riemannian version of the classical Newton's method after a finite number of iterations, consequently its convergence rate is superlinear/quadratic. Even at an early stage of development, we can observe from numerical experiments that DNM presented promising results when compared with the well known BFGS and Trust Regions methods. Moreover, damped Newton's method present better performance than the Newton's method in number of iteration and computational time.A note on three-step iterative method with seventh order of convergence for solving nonlinear equations.https://www.zbmath.org/1453.651022021-02-27T13:50:00+00:00"Srisarakham, Napassanan"https://www.zbmath.org/authors/?q=ai:srisarakham.napassanan"Thongmoon, Montri"https://www.zbmath.org/authors/?q=ai:thongmoon.montriSummary: In this paper, we present a new higher order iterative method for solving nonlinear equations. This method based on a Halley's iterative method and using predictor-corrector technique. The convergence analysis of this method is discussed. It is established that the new method has convergence order seven. Numerical tests show that the new method is comparable with the well-known existing methods and gives better results.An introduction to sequential Monte Carlo.https://www.zbmath.org/1453.620052021-02-27T13:50:00+00:00"Chopin, Nicolas"https://www.zbmath.org/authors/?q=ai:chopin.nicolas"Papaspiliopoulos, Omiros"https://www.zbmath.org/authors/?q=ai:papaspiliopoulos.omirosPublisher's description: This book provides a general introduction to sequential Monte Carlo (SMC) methods, also known as particle filters. These methods have become a staple for the sequential analysis of data in such diverse fields as signal processing, epidemiology, machine learning, population ecology, quantitative finance, and robotics.
The coverage is comprehensive, ranging from the underlying theory to computational implementation, methodology, and diverse applications in various areas of science. This is achieved by describing SMC algorithms as particular cases of a general framework, which involves concepts such as Feynman-Kac distributions, and tools such as importance sampling and resampling. This general framework is used consistently throughout the book.
Extensive coverage is provided on sequential learning (filtering, smoothing) of state-space (hidden Markov) models, as this remains an important application of SMC methods. More recent applications, such as parameter estimation of these models (through e.g. particle Markov chain Monte Carlo techniques) and the simulation of challenging probability distributions (in e.g. Bayesian inference or rare-event problems), are also discussed.
The book may be used either as a graduate text on Sequential Monte Carlo methods and state-space modeling, or as a general reference work on the area. Each chapter includes a set of exercises for self-study, a comprehensive bibliography, and a ``Python corner,'' which discusses the practical implementation of the methods covered. In addition, the book comes with an open source Python library, which implements all the algorithms described in the book, and contains all the programs that were used to perform the numerical experiments.Identification of physical processes via combined data-driven and data-assimilation methods.https://www.zbmath.org/1453.627972021-02-27T13:50:00+00:00"Chang, Haibin"https://www.zbmath.org/authors/?q=ai:chang.haibin"Zhang, Dongxiao"https://www.zbmath.org/authors/?q=ai:zhang.dongxiaoSummary: With the advent of modern data collection and storage technologies, data-driven approaches have been developed for discovering the governing partial differential equations (PDE) of physical problems. However, in the extant works the model parameters in the equations are either assumed to be known or have a linear dependency. Therefore, most of the realistic physical processes cannot be identified with the current data-driven PDE discovery approaches. In this study, an innovative framework is developed that combines data-driven and data-assimilation methods for simultaneously identifying physical processes and inferring model parameters. Spatiotemporal measurement data are first divided into a training data set and a testing data set. Using the training data set, a data-driven method is developed to learn the governing equation of the considered physical problem by identifying the occurred (or dominated) processes and selecting the proper empirical model. Through introducing a prediction error of the learned governing equation for the testing data set, a data-assimilation method is devised to estimate the uncertain model parameters of the selected empirical model. For the contaminant solute transport problem investigated, the results demonstrate that the proposed method can adequately identify the considered physical processes via concurrently discovering the corresponding governing equations and inferring uncertain parameters of nonlinear models, even in the presence of measurement errors. This work helps to broaden the applicable area of the research of data driven discovery of governing equations of physical problems.A new type of third-order finite volume multi-resolution WENO schemes on tetrahedral meshes.https://www.zbmath.org/1453.652662021-02-27T13:50:00+00:00"Zhu, Jun"https://www.zbmath.org/authors/?q=ai:zhu.jun"Shu, Chi-Wang"https://www.zbmath.org/authors/?q=ai:shu.chi-wangSummary: In this continuing paper of \textit{J. Zhu} and \textit{C.-W. Shu} [J. Comput. Phys. 375, 659--683 (2018; Zbl 1416.65286); ibid. 392, 19--33 (2019; Zbl 1452.76143)], we design a new third-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws on tetrahedral meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation. Comparing with classical third-order finite volume WENO schemes [\textit{Y.-T. Zhang} and \textit{C.-W. Shu}, Commun. Comput. Phys. 5, No. 2--4, 836--848 (2009; Zbl 1364.65177)] on tetrahedral meshes, the crucial advantages of such new multi-resolution WENO schemes are their simplicity and compactness with the application of only three unequal-sized central stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, their easy choice of arbitrary positive linear weights without considering the topology of the tetrahedral meshes, their optimal order of accuracy in smooth regions, and their suppression of spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO scheme can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite volume WENO scheme on tetrahedral meshes. By performing such new spatial reconstruction procedures and adopting a third-order TVD Runge-Kutta method for time discretization, the occupied memory is decreased and the computing efficiency is increased. This new third-order finite volume multi-resolution WENO scheme is suitable for large scale engineering applications and could maintain good convergence property for steady-state problems on tetrahedral meshes. Benchmark examples are computed to demonstrate the robustness and good performance of these new finite volume WENO schemes.Optimization of Steklov-Neumann eigenvalues.https://www.zbmath.org/1453.350562021-02-27T13:50:00+00:00"Ammari, Habib"https://www.zbmath.org/authors/?q=ai:ammari.habib-m"Imeri, Kthim"https://www.zbmath.org/authors/?q=ai:imeri.kthim"Nigam, Nilima"https://www.zbmath.org/authors/?q=ai:nigam.nilima|nigam.nilima-aSummary: This paper examines the Laplace equation with mixed boundary conditions, the Neumann and Steklov boundary conditions. This models a container with holes in it, like a pond filled with water but partly covered by immovable pieces on the surface. The main objective is to determine the right extent of the covering pieces, so that any shock inside the container yields a resonance. To this end, an algorithm is developed which uses asymptotic formulas concerning perturbations of the partitioning of the boundary pieces. Proofs for these formulas are established. Furthermore, this paper displays some results concerning bounds and examples with regards to the governing problem.Modified immersed boundary method for flows over randomly rough surfaces.https://www.zbmath.org/1453.760352021-02-27T13:50:00+00:00"Kwon, Chunsong"https://www.zbmath.org/authors/?q=ai:kwon.chunsong"Tartakovsky, Daniel M."https://www.zbmath.org/authors/?q=ai:tartakovsky.daniel-mSummary: Many phenomena, ranging from biology to electronics, involve flow over rough or irregular surfaces. We treat such surfaces as random fields and use an immersed boundary method (IBM) with discrete (random) forcing to solve resulting stochastic flow problems. Our approach relies on the Uhlmann formulation of the fluid-solid interaction force; computational savings stem from the modification of the time advancement scheme that obviates the need to solve the Poisson equation for pressure at each sub-step. We start by testing the proposed algorithm on two classical benchmark problems. The first deals with the Wannier problem of Stokesian flow around a cylinder in the vicinity of a moving plate. The second problem considers steady-state and transient flows over a stationary circular cylinder. Our simulation results show that our algorithm achieves second-order temporal accuracy. It is faster than the original IBM, while yielding consistent estimates of such quantities of interest as the drag and lift coefficients, the length of a recirculation zone in a cylinder's wake, and the Strouhal number. Then we use the proposed IBM algorithm to model flow over cylinders whose surface is either (deterministically) corrugated or (randomly) rough.A fully implicit mimetic finite difference scheme for general purpose subsurface reservoir simulation with full tensor permeability.https://www.zbmath.org/1453.761202021-02-27T13:50:00+00:00"Abushaikha, Ahmad S."https://www.zbmath.org/authors/?q=ai:abushaikha.ahmad-s"Terekhov, Kirill M."https://www.zbmath.org/authors/?q=ai:terekhov.kirill-mSummary: In the previous article [\textit{A. S. Abushaikha} et al., J. Comput. Phys. 346, 514--538 (2017; Zbl 1378.76042)], we presented a fully-implicit mixed hybrid finite element (MHFE) method for general-purpose compositional reservoir simulation. The present work extends the implementation for mimetic finite difference (MFD) discretization method. The new approach admits fully implicit solution on general polyhedral grids. The scheme couples the momentum and mass balance equations to assure conservation and applies a cubic equation-of-state for the fluid system. The flux conservativity is strongly imposed for the fully implicit approach and the Newton-Raphson method is used to linearize the system. We test the method through extensive numerical examples to demonstrate the convergence and accuracy on various shapes of polyhedral. We also compare the method to other discretization schemes for unstructured meshes and tensor permeability. Finally, we apply the method through applied computational cases to illustrate its robustness for full tensor anisotropic, highly heterogeneous and faulted reservoirs using unstructured grids.Adaptive mesh refinement in the fast lane.https://www.zbmath.org/1453.652992021-02-27T13:50:00+00:00"Dunning, D."https://www.zbmath.org/authors/?q=ai:dunning.d"Marts, W."https://www.zbmath.org/authors/?q=ai:marts.w"Robey, R. W."https://www.zbmath.org/authors/?q=ai:robey.robert-w"Bridges, P."https://www.zbmath.org/authors/?q=ai:bridges.pSummary: This paper presents an approach for constructing an adaptive mesh refinement (AMR) scheme, targeting next-generation computing hardware. The key to the design is the particular combination of aspects of cell-based AMR and patch-based AMR. We examine the feasibility of this new method with respect to correctness, preservation of circular symmetry, ease of programming and performance impacts on runtime and memory usage. This method exploration is done in CLAMR, a cell-based AMR mini-app that already runs on GPUs and other next-generation hardware platforms. The composability of the application is improved by decoupling the physics code and mesh code. Each level of the mesh is made independent through the use of \textit{phantom} cells. The net result is a clear pathway to getting the full application on the GPU while also minimizing development requirements to convert a regular grid application to AMR.A space-time adaptive finite element method with exponential time integrator for the phase field model of pitting corrosion.https://www.zbmath.org/1453.653172021-02-27T13:50:00+00:00"Gao, Huadong"https://www.zbmath.org/authors/?q=ai:gao.huadong"Ju, Lili"https://www.zbmath.org/authors/?q=ai:ju.lili"Li, Xiao"https://www.zbmath.org/authors/?q=ai:li.xiao"Duddu, Ravindra"https://www.zbmath.org/authors/?q=ai:duddu.ravindraSummary: In this paper we propose a space-time adaptive finite element method for the phase field model of pitting corrosion, which is a parabolic partial differential equation system consisting of a phase variable and a concentration variable. A major challenge in solving this phase field model is that the problem is very stiff, which makes the time step size extremely small for standard temporal discretizations. Another difficulty is that a high spatial resolution is required to capture the steep gradients within the diffused interface, which results in very large number of degrees of freedom for uniform meshes. To overcome the stiffness of this model, we combine the Rosenbrock-Euler exponential integrator with Crank-Nicolson scheme for the temporal discretization. Moreover, by exploiting the fact that the speed of the corroding interface decreases with time, we derive an adaptive time stepping formula. For the spatial approximation, we propose a simple and efficient strategy to generate adaptive meshes that reduces the computational cost significantly. Thus, the proposed method utilizes local adaptivity and mesh refinement for efficient simulation of the corrosive dissolution over long times in heterogeneous media with complex microstructures. We also present an extensive set of numerical experiments in both two and three dimensional spaces to demonstrate efficiency and robustness of the proposed method.Implicit method for the solution of supersonic and hypersonic 3D flow problems with lower-upper symmetric-Gauss-Seidel preconditioner on multiple graphics processing units.https://www.zbmath.org/1453.760892021-02-27T13:50:00+00:00"Bocharov, A. N."https://www.zbmath.org/authors/?q=ai:bocharov.a-n"Evstigneev, N. M."https://www.zbmath.org/authors/?q=ai:evstigneev.nikolai-m"Petrovskiy, V. P."https://www.zbmath.org/authors/?q=ai:petrovskiy.v-p"Ryabkov, O. I."https://www.zbmath.org/authors/?q=ai:ryabkov.o-i"Teplyakov, I. O."https://www.zbmath.org/authors/?q=ai:teplyakov.i-oSummary: The paper describes a numerical method for the solution of stationary gas dynamics 3D spatial equations on unstructured grids that is designed for multiple graphics processing unit (GPU) computational architecture. Discretization of governing equations is done using first and second order TVD schemes. The Newton's method with simple pseudo time-step homotopy is used to solve the problem. Each iteration step involves solution of the linear system originated from the linearization of gas dynamics equations. Krylov subspace iterative methods are used to solve the linear system. The main aim of the paper is to describe a preconditioning Lower-Upper Symmetric-Gauss-Seidel (LU-SGS) method and its adaptation on multiple GPU computational systems. It is shown that deliberately reordered matrices with rearranged solution process of arising lower and upper triangular linear systems allow one to obtain close algebraic properties to the original single threaded LU-SGS. The method is benchmarked against published results. The analysis of computational efficiency and acceleration is presented for different flows with Mach number ranging from 1.2 up to 25.A two-layer shallow flow model with two axes of integration, well-balanced discretization and application to submarine avalanches.https://www.zbmath.org/1453.652462021-02-27T13:50:00+00:00"Delgado-Sánchez, J. M."https://www.zbmath.org/authors/?q=ai:delgado-sanchez.juan-m"Bouchut, Francois"https://www.zbmath.org/authors/?q=ai:bouchut.francois"Fernández-Nieto, E. D."https://www.zbmath.org/authors/?q=ai:fernandez-nieto.enrique-domingo"Mangeney, A."https://www.zbmath.org/authors/?q=ai:mangeney.anne"Narbona-Reina, G."https://www.zbmath.org/authors/?q=ai:narbona-reina.gladysSummary: We propose a two-layer model with two different axes of integration and a well-balanced finite volume method. The purpose is to study submarine avalanches and generated tsunamis by a depth-averaged model with different averaged directions for the fluid and the granular layers. Two-layer shallow depth-averaged models usually consider either Cartesian or local coordinates for both layers. However, the motion characteristics of the granular layer and the water wave are different: the granular flow velocity is mainly oriented downslope while water motion related to tsunami wave propagation is mostly horizontal. As a result, the shallow approximation and depth-averaging have to be imposed (i) in the direction normal to the topography for the granular flow and (ii) in the vertical direction for the water layer. To deal with this problem, we define a reference plane related to topography variations and use the associated local coordinates to derive the granular layer equations whereas Cartesian coordinates are used for the fluid layer. Depth-averaging is done orthogonally to that reference plane for the granular layer equations and in the vertical direction for the fluid layer equations. Then, a finite volume method is defined based on an extension of the hydrostatic reconstruction. The proposed method is exactly well-balanced for two kinds of stationary solutions: the classical one, when both water and granular masses are at rest; the second one, when only the granular mass is at rest. Several tests are presented to get insight into the sensitivity of the granular flow, deposit and generated water waves to the choice of the coordinate systems. Our results show that even for moderate slopes (up to 30\degree), strong relative errors on the avalanche dynamics and deposit (up to 60\%) and on the generated water waves (up to 120\%) are made when using Cartesian coordinates for both layers instead of an appropriate local coordinate system as proposed here.A discontinuous Galerkin method for the Aw-Rascle traffic flow model on networks.https://www.zbmath.org/1453.653102021-02-27T13:50:00+00:00"Buli, Joshua"https://www.zbmath.org/authors/?q=ai:buli.joshua"Xing, Yulong"https://www.zbmath.org/authors/?q=ai:xing.yulongSummary: Macroscopic models for flows strive to depict the physical world by considering quantities of interest at the aggregate level versus focusing on each discrete particle in the system. Many practical problems of interest such as the blood flow in the circulatory system, irrigation channels, supply chains, and vehicular traffic on freeway systems can all be modeled using hyperbolic conservation laws that track macroscopic quantities through a network. In this paper we consider the latter, specifically the second-order Aw-Rascle (AR) traffic flow model on a network, and propose a discontinuous Galerkin (DG) method for solving the AR system of hyperbolic partial differential equations with appropriate coupling conditions at the junctions. On each road, the standard DG method with Lax-Friedrichs flux is employed, and at the junction, we solve an optimization problem to evaluate the numerical flux of the DG method. As the choice of well-posed coupling conditions for the AR model is not unique, we test different coupling conditions at the junctions. Numerical examples are provided to demonstrate the high-order accuracy, and comparison of results between the first-order Lighthill-Whitham-Richards model and the second-order AR model. The ability of the model to capture the capacity drop phenomenon is also explored.Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries.https://www.zbmath.org/1453.653062021-02-27T13:50:00+00:00"Yu, Fei"https://www.zbmath.org/authors/?q=ai:yu.fei"Guo, Zhenlin"https://www.zbmath.org/authors/?q=ai:guo.zhenlin"Lowengrub, John"https://www.zbmath.org/authors/?q=ai:lowengrub.john-sSummary: The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a larger, regular domain. The original PDE is reformulated using a smoothed characteristic function of the complex domain and source terms are introduced to approximate the boundary conditions. The reformulated equation, which is independent of the dimension and domain geometry, can be solved by standard numerical methods and the same solver can be used for any domain geometry. A challenge is making the method higher-order accurate. For Dirichlet boundary conditions, which we focus on here, current implementations demonstrate a wide range in their accuracy but generally the methods yield at best first order accuracy in \(\epsilon\), the parameter that characterizes the width of the region over which the characteristic function is smoothed. Typically, \(\epsilon\propto h\), the grid size. Here, we analyze the diffuse-domain PDEs using matched asymptotic expansions and explain the observed behaviors. Our analysis also identifies simple modifications to the diffuse-domain PDEs that yield higher-order accuracy in \(\epsilon\), e.g., \(O(\epsilon^2)\) in the \(L^2\) norm and \(O(\epsilon^p)\) with \(1.5\leq p \leq 2\) in the \(L^\infty\) norm. Our analytic results are confirmed numerically in stationary and moving domains where the level set method is used to capture the dynamics of the domain boundary and to construct the smoothed characteristic function.Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up.https://www.zbmath.org/1453.651182021-02-27T13:50:00+00:00"Holm, Bärbel"https://www.zbmath.org/authors/?q=ai:holm.barbel"Wihler, Thomas P."https://www.zbmath.org/authors/?q=ai:wihler.thomas-pascalAuthors' abstract: We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.
Reviewer: Mikhail Yu. Kokurin (Yoshkar-Ola)The image-based multiscale multigrid solver, preconditioner, and reduced order model.https://www.zbmath.org/1453.654312021-02-27T13:50:00+00:00"Yushu, Dewen"https://www.zbmath.org/authors/?q=ai:yushu.dewen"Matouš, Karel"https://www.zbmath.org/authors/?q=ai:matous.karelSummary: We present a novel image-based multiscale multigrid solver that can efficiently address the computational complexity associated with highly heterogeneous systems. This solver is developed based on an image-based, multiresolution model that enables reliable data flow between corresponding computational grids and provides large data compression. A set of inter-grid operators is constructed based on the microstructural data which remedies the issue of missing coarse grid information. Moreover, we develop an image-based multiscale preconditioner from the multiscale coarse images which does not traverse through any intermediate grid levels and thus leads to a faster solution process. Finally, an image-based reduced order model is designed by prolongating the coarse-scale solution to approximate the fine-scale one with improved accuracy. The numerical robustness and efficiency of this image-based computational framework is demonstrated on a two-dimensional example with high degrees of data heterogeneity and geometrical complexity.Estimating and localizing the algebraic and total numerical errors using flux reconstructions.https://www.zbmath.org/1453.654142021-02-27T13:50:00+00:00"Papež, J."https://www.zbmath.org/authors/?q=ai:papez.jan"Strakoš, Z."https://www.zbmath.org/authors/?q=ai:strakos.zdenek"Vohralík, M."https://www.zbmath.org/authors/?q=ai:vohralik.martinSummary: This paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization of the Poisson model problem and an arbitrary iterative algebraic solver. The derived bounds do not contain any unspecified constants and allow estimating the local distribution of both errors over the computational domain. Combining these bounds, we also obtain guaranteed upper and lower bounds on the discretization error. This allows to propose novel mathematically justified stopping criteria for iterative algebraic solvers ensuring that the algebraic error will lie below the discretization one. Our upper algebraic and total error bounds are based on locally reconstructed fluxes in \({\mathbf {H}}(\operatorname{div},\varOmega )\), whereas the lower algebraic and total error bounds rely on locally constructed \(H^1_0(\varOmega )\)-liftings of the algebraic and total residuals. We prove global and local efficiency of the upper bound on the total error and its robustness with respect to the approximation polynomial degree. Relationships to the previously published estimates on the algebraic error are discussed. Theoretical results are illustrated on numerical experiments for higher-order finite element approximations and the preconditioned conjugate gradient method. They in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over the computational domain and illustrate the associated cost.Scalable angular adaptivity for Boltzmann transport.https://www.zbmath.org/1453.652972021-02-27T13:50:00+00:00"Dargaville, Steven"https://www.zbmath.org/authors/?q=ai:dargaville.steven"Buchan, A. G."https://www.zbmath.org/authors/?q=ai:buchan.andrew-g"Smedley-Stevenson, R. P."https://www.zbmath.org/authors/?q=ai:smedley-stevenson.richard-p"Smith, P. N."https://www.zbmath.org/authors/?q=ai:smith.paul-n"Pain, C. C."https://www.zbmath.org/authors/?q=ai:pain.christopher-cSummary: This paper describes an angular adaptivity algorithm for Boltzmann transport applications which for the first time shows evidence of \(\mathcal{O}(n)\) scaling in both runtime and memory usage, where \(n\) is the number of adapted angles. This adaptivity uses Haar wavelets, which perform structured \(h\)-adaptivity built on top of a hierarchical \(P^0\) FEM discretisation of a 2D angular domain, allowing different anisotropic angular resolution to be applied across space/energy. These wavelets can be mapped back to their underlying \(P^0\) space scalably, allowing traditional DG-sweep algorithms if desired. Instead we build a spatial discretisation on unstructured grids designed to use less memory than competing alternatives in general applications and construct a compatible matrix-free multigrid method which can handle our adapted angular discretisation. Fixed angular refinement, along with regular and goal-based error metrics are shown in three example problems taken from neutronics/radiative transfer applications.Coercing machine learning to output physically accurate results.https://www.zbmath.org/1453.681642021-02-27T13:50:00+00:00"Geng, Zhenglin"https://www.zbmath.org/authors/?q=ai:geng.zhenglin"Johnson, Daniel"https://www.zbmath.org/authors/?q=ai:johnson.daniel-cowan|johnson.daniel-d|johnson.daniel-h|johnson.daniel-p"Fedkiw, Ronald"https://www.zbmath.org/authors/?q=ai:fedkiw.ronald-pSummary: Many machine/deep learning artificial neural networks are trained to simply be interpolation functions that map input variables to output values interpolated from the training data in a linear/nonlinear fashion. Even when the input/output pairs of the training data are physically accurate (e.g. the results of an experiment or numerical simulation), interpolated quantities can deviate quite far from being physically accurate. Although one could project the output of a network into a physically feasible region, such a postprocess is not captured by the energy function minimized when training the network; thus, the final projected result could incorrectly deviate quite far from the training data. We propose folding any such projection or postprocess directly into the network so that the final result is correctly compared to the training data by the energy function. Although we propose a general approach, we illustrate its efficacy on a specific convolutional neural network that takes in human pose parameters (joint rotations) and outputs a prediction of vertex positions representing a triangulated cloth mesh. While the original network outputs vertex positions with erroneously high stretching and compression energies, the new network trained with our physics ``prior'' remedies these issues producing highly improved results.A Bayesian structural-change analysis via the stochastic approximation Monte Carlo and Gibbs sampler.https://www.zbmath.org/1453.620682021-02-27T13:50:00+00:00"Cheon, Sooyoung"https://www.zbmath.org/authors/?q=ai:cheon.sooyoung"Kim, Jaehee"https://www.zbmath.org/authors/?q=ai:kim.jaehee-hSummary: In this article, we propose a Bayesian approach to estimate the multiple structural change-points in a level and the trend when the number of change-points is unknown. Our formulation of the structural-change model involves a binary discrete variable that indicates the structural change. The determination of the number and the form of structural changes are considered as a model selection issue in Bayesian structural-change analysis. We apply an advanced Monte Carlo algorithm, the stochastic approximation Monte Carlo (SAMC) algorithm, to this structural-change model selection issue. SAMC effectively functions for the complex structural-change model estimation, since it prevents entrapment in local posterior mode. The estimation of the model parameters in each regime is made using the Gibbs sampler after each change-point is detected. The performance of our proposed method has been investigated on simulated and real data sets, a long time series of US real gross domestic product, US uses of force between 1870 and 1994 and 1-year time series of temperature in Seoul, South Korea.Symmetric alternating direction method with indefinite proximal regularization for linearly constrained convex optimization.https://www.zbmath.org/1453.651332021-02-27T13:50:00+00:00"Gao, Bin"https://www.zbmath.org/authors/?q=ai:gao.bin"Ma, Feng"https://www.zbmath.org/authors/?q=ai:ma.fengThe authors confirm that the symmetric alternating direction method of multipliers can also be regularized with an indefinite proximal term. The global convergence is theoretically proved and its worst-case convergence rate in an ergodic sense is established. Numerical experiments are presented.
Reviewer: Hans Benker (Merseburg)The Monte-Carlo algorithm for the solving of systems of linear algebraic equations by the Seidel method.https://www.zbmath.org/1453.650752021-02-27T13:50:00+00:00"Tovstik, T. M."https://www.zbmath.org/authors/?q=ai:tovstik.tatiana-m|tovstik.tatyana-mikhailovna"Volosenko, K. S."https://www.zbmath.org/authors/?q=ai:volosenko.k-sSummary: The iteration algorithm is used to solve systems of linear algebraic equations by the Monte-Carlo method. Each next iteration is simulated as a random vector such that its expectation coincides with the Seidel approximation of the iteration process. We deduce a system of linear equations such that mutual correlations of components of the limit vector and correlations of two iterations satisfy them. We prove that limit dispersions of the random vector of solutions of the system exist and are finite.An energy stable \(C^0\) finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density.https://www.zbmath.org/1453.653432021-02-27T13:50:00+00:00"Shen, Lingyue"https://www.zbmath.org/authors/?q=ai:shen.lingyue"Huang, Huaxiong"https://www.zbmath.org/authors/?q=ai:huang.huaxiong"Lin, Ping"https://www.zbmath.org/authors/?q=ai:lin.ping"Song, Zilong"https://www.zbmath.org/authors/?q=ai:song.zilong"Xu, Shixin"https://www.zbmath.org/authors/?q=ai:xu.shixinSummary: In this paper, we focus on modeling and simulation of two-phase flow problems with moving contact lines and variable density. A thermodynamically consistent phase-field model with general Navier boundary condition is developed based on the concept of quasi-incompressibility and the energy variational method. A mass conserving \(C^0\) finite element scheme is proposed to solve the PDE system. Energy stability is achieved at the fully discrete level. Various numerical results confirm that the proposed scheme for both \(P^1\) element and \(P^2\) element are energy stable.Solving spatial-fractional partial differential diffusion equations by spectral method.https://www.zbmath.org/1453.654602021-02-27T13:50:00+00:00"Nie, Ningming"https://www.zbmath.org/authors/?q=ai:nie.ningming"Huang, Jianfei"https://www.zbmath.org/authors/?q=ai:huang.jianfei"Wang, Wenjia"https://www.zbmath.org/authors/?q=ai:wang.wenjia"Tang, Yifa"https://www.zbmath.org/authors/?q=ai:tang.yifaSummary: This paper focuses on numerical solution of an initial-boundary value problem of spatial-fractional partial differential diffusion equation. The proposed numerical method is based on Legendre spectral method for Riemann-Liouville fractional derivative in space and a finite difference scheme in time. Numerical analysis of stability and convergence for our method is established rigourously. Finally, numerical results verify the validity of the theoretical analysis.Harnack inequalities for stochastic heat equation with locally unbounded drift.https://www.zbmath.org/1453.601262021-02-27T13:50:00+00:00"Yin, Xiuwei"https://www.zbmath.org/authors/?q=ai:yin.xiuwei"Shen, Guangjun"https://www.zbmath.org/authors/?q=ai:shen.guangjun"Zhang, Jinhong"https://www.zbmath.org/authors/?q=ai:zhang.jinhongThis paper is concerned with the Harnack inequalities for the stochastic heat equation with the Neumann boundary condition and the additive white noise in one dimension. For the corresponding Markov operator, the authors first prove the Harnack inequality, based on the coupling by change of measure and Krylov's estimate. Then they prove the shift (log-)Harnack inequality. Moreover, the application to the equivalence between the corresponding distributions of solutions is also obtained.
Reviewer: Deng Zhang (Shanghai)Well balanced residual distribution for the ALE spherical shallow water equations on moving adaptive meshes.https://www.zbmath.org/1453.652952021-02-27T13:50:00+00:00"Arpaia, Luca"https://www.zbmath.org/authors/?q=ai:arpaia.luca"Ricchiuto, Mario"https://www.zbmath.org/authors/?q=ai:ricchiuto.marioSummary: We consider the numerical approximation of the Shallow Water Equations (SWEs) in spherical geometry for oceanographic applications. To provide enhanced resolution of moving fronts present in the flow we consider adaptive discrete approximations on moving triangulations of the sphere. To this end, we re-state all Arbitrary Lagrangian Eulerian (ALE) transport formulas, as well as the volume transformation laws, for a 2D manifold. Using these results, we write the set of ALE-SWEs on the sphere. We then propose a Residual Distribution discrete approximation of the governing equations. Classical properties as the DGCL and the C-property (well balancedness) are reformulated in this more general context. An adaptive mesh movement strategy is proposed. The discrete framework obtained is thoroughly tested on standard benchmarks in large scale oceanography to prove their potential as well as the advantage brought by the adaptive mesh movement.Numerical simulations of one laser-plasma model based on Poisson structure.https://www.zbmath.org/1453.652572021-02-27T13:50:00+00:00"Li, Yingzhe"https://www.zbmath.org/authors/?q=ai:li.yingzhe"Sun, Yajuan"https://www.zbmath.org/authors/?q=ai:sun.yajuan"Crouseilles, Nicolas"https://www.zbmath.org/authors/?q=ai:crouseilles.nicolasSummary: In this paper, a bracket structure is proposed for the laser-plasma interaction model introduced in [\textit{A. Ghizzo} et al., J. Comput. Phys. 90, No. 2, 431--457 (1990; Zbl 0702.76080)], and it is proved by direct calculations that the bracket is Poisson which satisfies the Jacobi identity. Then splitting methods in time are proposed based on the Poisson structure. For the quasi-relativistic case, the Hamiltonian splitting leads to three subsystems which can be solved exactly. The conservative splitting is proposed for the fully relativistic case, and three one-dimensional conservative subsystems are obtained. Combined with the splittings in time, in phase space discretization we use the Fourier spectral and finite volume methods. It is proved that the discrete charge and discrete Poisson equation are conserved by our numerical schemes. Numerically, some numerical experiments are conducted to verify good conservations for the charge, energy and Poisson equation.A phase-field moving contact line model with soluble surfactants.https://www.zbmath.org/1453.761462021-02-27T13:50:00+00:00"Zhu, Guangpu"https://www.zbmath.org/authors/?q=ai:zhu.guangpu"Kou, Jisheng"https://www.zbmath.org/authors/?q=ai:kou.jisheng"Yao, Jun"https://www.zbmath.org/authors/?q=ai:yao.jun"Li, Aifen"https://www.zbmath.org/authors/?q=ai:li.aifen"Sun, Shuyu"https://www.zbmath.org/authors/?q=ai:sun.shuyuSummary: A phase-field moving contact line model is presented for a two-phase system with soluble surfactants. With the introduction of some scalar auxiliary variables, the original free energy functional is transformed into an equivalent form, and then a new governing system is obtained. The resulting model consists of two Cahn-Hilliard-type equations and incompressible Navier-Stokes equation with variable densities, together with the generalized Navier boundary condition for the moving contact line. We prove that the proposed model satisfies the total energy dissipation with time. To numerically solve such a complex system, we develop a nonlinearly coupled scheme with unconditional energy stability. A splitting method based on pressure stabilization is used to solve the Navier-Stokes equation. Some subtle implicit-explicit treatments are adopted to discretize convection and stress terms. A stabilization term is artificially added to balance the explicit nonlinear term associated with the surface energy at the fluid-solid interface. We rigorously prove that the proposed scheme can preserve the discrete energy dissipation. An efficient finite difference method on staggered grids is used for the spatial discretization. Numerical results in both two and three dimensions demonstrate the accuracy and energy stability of the proposed scheme. Using our model and numerical scheme, we investigate the wetting behavior of droplets on a solid wall. Numerical results indicate that surfactants can affect the wetting properties of droplet by altering the value of contact angles.Evaluation of Abramowitz functions in the right half of the complex plane.https://www.zbmath.org/1453.650482021-02-27T13:50:00+00:00"Gimbutas, Zydrunas"https://www.zbmath.org/authors/?q=ai:gimbutas.zydrunas"Jiang, Shidong"https://www.zbmath.org/authors/?q=ai:jiang.shidong"Luo, Li-Shi"https://www.zbmath.org/authors/?q=ai:luo.lishiSummary: A numerical scheme is developed for the evaluation of Abramowitz functions \(J_n\) in the right half of the complex plane. For \(n=-1,\dots,2\), the scheme utilizes series expansions for \(|z|<1\), asymptotic expansions for \(|z|>R\) with \(R\) determined by the required precision, and least squares Laurent polynomial approximations on each sub-region in the intermediate region \(1\leq |z|\leq R\). For \(n>2\), \(J_n\) is evaluated via a forward recurrence relation. The scheme achieves nearly machine precision for \(n=-1,\dots,2\) at a cost that is competitive as compared with software packages for the evaluation of other special functions in the complex domain.Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations.https://www.zbmath.org/1453.653452021-02-27T13:50:00+00:00"Torres-Sánchez, Alejandro"https://www.zbmath.org/authors/?q=ai:torres-sanchez.alejandro"Santos-Oliván, Daniel"https://www.zbmath.org/authors/?q=ai:santos-olivan.daniel"Arroyo, Marino"https://www.zbmath.org/authors/?q=ai:arroyo.marinoSummary: We introduce a new method, the Local Monge Parametrizations (LMP) method, to approximate tensor fields on general surfaces given by a collection of local parametrizations, e.g. as in finite element or NURBS surface representations. Our goal is to use this method to solve numerically tensor-valued partial differential equations (PDEs) on surfaces. Previous methods use scalar potentials to numerically describe vector fields on surfaces, at the expense of requiring higher-order derivatives of the approximated fields and limited to simply connected surfaces, or represent tangential tensor fields as tensor fields in 3D subjected to constraints, thus increasing the essential number of degrees of freedom. In contrast, the LMP method uses an optimal number of degrees of freedom to represent a tensor, is general with regards to the topology of the surface, and does not increase the order of the PDEs governing the tensor fields. The main idea is to construct maps between the element parametrizations and a local Monge parametrization around each node. We test the LMP method by approximating in a least-squares sense different vector and tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply the LMP method to two physical models on surfaces, involving a tension-driven flow (vector-valued PDE) and nematic ordering (tensor-valued PDE), on different topologies. The LMP method thus solves the long-standing problem of the interpolation of tensors on general surfaces with an optimal number of degrees of freedom.Kernel-based collocation methods for heat transport on evolving surfaces.https://www.zbmath.org/1453.653562021-02-27T13:50:00+00:00"Chen, Meng"https://www.zbmath.org/authors/?q=ai:chen.meng.1|chen.meng"Ling, Leevan"https://www.zbmath.org/authors/?q=ai:ling.leevanSummary: We propose algorithms for solving convective-diffusion partial differential equations (PDEs), which model surfactant concentration and heat transport on evolving surfaces, based on extrinsic kernel-based meshless collocation methods. The algorithms can be classified into two categories: one collocates PDEs extrinsically and analytically, and the other approximates surface differential operators by meshless pseudospectral approaches. The former is specifically designed to handle PDEs on evolving surfaces defined by parametric equations, and the latter works on surface evolutions based on point clouds. After some convergence studies and comparisons, we demonstrate that the proposed method can solve challenging PDEs posed on surfaces with high curvatures with discontinuous initial conditions with correct physics.An Eulerian-Lagrangian-Lagrangian method for solving fluid-structure interaction problems with bulk solids.https://www.zbmath.org/1453.740282021-02-27T13:50:00+00:00"Han, Dong"https://www.zbmath.org/authors/?q=ai:han.dong"Liu, G. R."https://www.zbmath.org/authors/?q=ai:liu.gui-rong"Abdallah, Shaaban"https://www.zbmath.org/authors/?q=ai:abdallah.shaabanSummary: This paper presents an Eulerian-Lagrangian-Lagrangian (ELL) method to solve the fluid-structure interaction (FSI) problem, in which a bulk solid is immersed in the fluid. The ELL approach uses a small portion of the Lagrangian fluid to wrap the solid body, and they are treated as an enlarge``composite solid''. Then the conventional immersed finite element method (IFEM) is implemented. The velocity condition of the moving interface is explicitly imposed within the unified Lagrangian grids of the composite solid domain, and then smoothly transferred onto the Eulerian fluid nodes nearby the FSI interaction zone. The inaccurate numerical FSI interface in IFEM is hence significantly improved. The idea of warping Lagrangian fluid makes ELL a combination of the body-conforming method and the immersed type method. It is straightforward but offers an excellent physical explanation that makes ELL more realistic. Several numerical examples are computed to demonstrate the superior performance of the proposed approach for solving FSI problems with bulk solids, in comparison with the conventional IFEM.High order efficient splittings for the semiclassical time-dependent Schrödinger equation.https://www.zbmath.org/1453.810302021-02-27T13:50:00+00:00"Blanes, Sergio"https://www.zbmath.org/authors/?q=ai:blanes.sergio"Gradinaru, Vasile"https://www.zbmath.org/authors/?q=ai:gradinaru.vasileSummary: Standard numerical schemes with time-step \(h\) deteriorate (e.g. like \(\varepsilon^{-2}h^2)\) in the presence of a small semiclassical parameter \(\epsilon\) in the time-dependent Schrödinger equation. The recently introduced semiclassical splitting was shown to be of order \(\mathcal{O}(\varepsilon h^2)\). We present now an algorithm that is of order \(\mathcal{O}(\varepsilon h^7+\varepsilon^2 h^6+\varepsilon^3h^4)\) at the expense of roughly three times the computational effort of the semiclassical splitting and another that is of order \(\mathcal{O}(\varepsilon h^6+\varepsilon^2h^4)\) at the \textit{same} expense of the computational effort of the semiclassical splitting.A stable parareal-like method for the second order wave equation.https://www.zbmath.org/1453.652752021-02-27T13:50:00+00:00"Nguyen, Hieu"https://www.zbmath.org/authors/?q=ai:nguyen.hieu-tat|nguyen.hieu-van|nguyen.hieu-duy|nguyen-trung-hieu.|nguyen-hieu-duc.|nguyen.hieu-minh"Tsai, Richard"https://www.zbmath.org/authors/?q=ai:tsai.yen-hsi-richardSummary: A new parallel-in-time iterative method is proposed for solving the homogeneous second-order wave equation. The new method involves a coarse scale propagator, allowing for larger time steps, and a fine scale propagator which fully resolves the medium using finer spatial grid and uses shorter time steps. The fine scale propagator is run in parallel for short time intervals. The two propagators are coupled in an iterative way that resembles the standard parareal method [\textit{J.-L. Lions} et al., C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 7, 661--668 (2001; Zbl 0984.65085)]. We present a data-driven strategy in which the computed data gathered from each iteration are re-used to stabilize the coupling by minimizing the wave energy residual of the fine and coarse propagated solutions. Several examples, including a wave speed with discontinuities, are provided to demonstrate the effectiveness of the proposed method.Numerical approximation of the Schrödinger equation with concentrated potential.https://www.zbmath.org/1453.653762021-02-27T13:50:00+00:00"Banjai, L."https://www.zbmath.org/authors/?q=ai:banjai.lehel|banyai.ladislaus-alexander"López-Fernández, Maria"https://www.zbmath.org/authors/?q=ai:lopez-fernandez.mariaSummary: We present a family of algorithms for the numerical approximation of the Schrödinger equation with potential concentrated at a finite set of points. Our methods belong to the so-called fast and oblivious convolution quadrature algorithms. These algorithms are special implementations of Lubich's Convolution Quadrature which allow, for certain applications in particular parabolic problems, to significantly reduce the computational cost and memory requirements. Recently it has been noticed that their use can be extended to some hyperbolic problems. Here we propose a new family of such efficient algorithms tailored to the features of the Green's function for Schrödinger equations. In this way, we are able to keep the computational cost and the storage requirements significantly below existing approaches. These features allow us to perform reliable numerical simulations for longer times even in cases where the solution becomes highly oscillatory or seems to develop finite time blow-up. We illustrate our new algorithm with several numerical experiments.Development and analysis of both finite element and fourth-order in space finite difference methods for an equivalent Berenger's PML model.https://www.zbmath.org/1453.780092021-02-27T13:50:00+00:00"Huang, Yunqing"https://www.zbmath.org/authors/?q=ai:huang.yunqing"Chen, Min"https://www.zbmath.org/authors/?q=ai:chen.min.3|chen.min|chen.min.1|chen.min.2"Li, Jichun"https://www.zbmath.org/authors/?q=ai:li.jichunSummary: This paper deals with an equivalent Berenger's Perfectly Matched Layer (PML) model. We first develop a finite element scheme using edge elements to solve this model. We prove a discrete stability of this method, which inherits the stability obtained in the continuous case. Then we propose a fourth-order in space finite difference scheme for solving this PML model. Numerical stability similar to the continuous stability and the optimal error estimate are established for the difference scheme. Here only second order time discretizations are considered for both schemes. Finally, numerical results are presented to justify our analysis and demonstrate the effectiveness of this PML model for absorbing impinging waves.A mapping-function-free WENO-M scheme with low computational cost.https://www.zbmath.org/1453.652192021-02-27T13:50:00+00:00"Hong, Zheng"https://www.zbmath.org/authors/?q=ai:hong.zheng"Ye, Zhengyin"https://www.zbmath.org/authors/?q=ai:ye.zhengyin"Meng, Xianzong"https://www.zbmath.org/authors/?q=ai:meng.xianzongSummary: The classical WENO-JS scheme developed by \textit{G.-S. Jiang} and \textit{C.-W. Shu} [J. Comput. Phys. 126, No. 1, 202--228 (1996; Zbl 0877.65065)] suffers accuracy loss near a critical point in smooth region. An extra mapping process was introduced by \textit{A. K. Henrick} et al. [J. Comput. Phys. 207, No. 2, 542--567 (2005; Zbl 1072.65114)] to overcome this problem with significantly increased computational cost, known as WENO-M scheme. In order to reduce the cost brought by the mapping process, a pre-discrete mapping method is proposed in this article. The cost of this new method is not only low, but also independent of the complexity of mapping function. Numerical experiments with one- and two- dimensional benchmark problems are conducted to demonstrate the effectiveness of this new method. Compared to the original method, the cost is reduced by 52\%.A positivity-preserving Lagrangian discontinuous Galerkin method for ideal magnetohydrodynamics equations in one-dimension.https://www.zbmath.org/1453.760872021-02-27T13:50:00+00:00"Zou, Shijun"https://www.zbmath.org/authors/?q=ai:zou.shijun"Yu, Xijun"https://www.zbmath.org/authors/?q=ai:yu.xijun"Dai, Zihuan"https://www.zbmath.org/authors/?q=ai:dai.zihuanSummary: In this paper, we propose a conservative Lagrangian Discontinuous Galerkin (DG) scheme for solving the ideal compressible magnetohydrodynamics (MHD) equations in one-dimensional. This scheme can preserve positivity of physically positive variables such as density and thermal pressure. We first develop a Lagrangian HLLD approximate Riemann solver which can keep positivity-preserving property under some appropriate signal speeds. With this solver a first order positivity-preserving Lagrangian DG scheme can be constructed. Then we design a high order positivity-preserving and conservative Lagrangian DG scheme by using the strong stability preserving (SSP) high order time discretizations and the positivity-preserving scaling limiter. We adopt TVB minmod limiter with the local characteristic fields for the ideal MHD system to control spurious oscillations around the shock wave. Some numerical examples are presented to demonstrate the accuracy and positivity-preserving property of our scheme.A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations.https://www.zbmath.org/1453.653702021-02-27T13:50:00+00:00"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui.5|zhang.hui.7|zhang.hui.2|zhang.hui.1|zhang.hui.11|zhang.hui|zhang.hui.6|zhang.hui.9|zhang.hui.3|zhang.hui.8|zhang.hui.10|zhang.hui.4"Jiang, Xiaoyun"https://www.zbmath.org/authors/?q=ai:jiang.xiaoyun"Zeng, Fanhai"https://www.zbmath.org/authors/?q=ai:zeng.fanhai"Karniadakis, George Em"https://www.zbmath.org/authors/?q=ai:karniadakis.george-emSummary: The reaction-diffusion model can generate a wide variety of spatial patterns, which has been widely applied in chemistry, biology, and physics, even used to explain self-regulated pattern formation in the developing animal embryo. In this work, a second-order stabilized semi-implicit time-stepping Fourier spectral method for the reaction-diffusion systems of equations with space described by the fractional Laplacian is developed. We adopt the temporal-spatial error splitting argument to illustrate that the proposed method is stable without imposing the CFL condition, and an optimal \(L^2\)-error estimate in space is proved. We also analyze the linear stability of the stabilized semi-implicit method and obtain a practical criterion to choose the time step size to guarantee the stability of the semi-implicit method. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen-Cahn, Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator, which are quite different from the patterns of the corresponding integer-order model.Improved \(\vartheta\)-methods for stochastic Volterra integral equations.https://www.zbmath.org/1453.650162021-02-27T13:50:00+00:00"Conte, Dajana"https://www.zbmath.org/authors/?q=ai:conte.dajana"D'Ambrosio, Raffaele"https://www.zbmath.org/authors/?q=ai:dambrosio.raffaele"Paternoster, Beatrice"https://www.zbmath.org/authors/?q=ai:paternoster.beatriceSummary: The paper introduces improved stochastic \(\vartheta\)-methods for the numerical integration of stochastic Volterra integral equations. Such methods, compared to those introduced by the authors in [Discrete Contin. Dyn. Syst., Ser. B 23, No. 7, 2695--2708 (2018; Zbl 1398.65348)], have better stability properties. This is here made possible by inheriting the stability properties of the corresponding methods for systems of stochastic differential equations. Such a superiority is confirmed by a comparison of the stability regions.A numerically stable spherical harmonics solution for the neutron transport equation in a spherical shell.https://www.zbmath.org/1453.654522021-02-27T13:50:00+00:00"Garcia, R. D. M."https://www.zbmath.org/authors/?q=ai:garcia.roberto-d-mSummary: A numerically stable version of the spherical harmonics \((P_N)\) method for solving the one-speed neutron transport equation with \(L\) th order anisotropic scattering in a spherical shell is developed. Implementing a stable \(P_N\) solution for this problem is a challenging task for which no satisfactory answer has been given in the literature. The approach used in this work follows and generalizes a previous work on a problem whose domain is defined by the exterior of a sphere. First, a transformation is used to reduce the original transport equation in spherical geometry to a plane-geometry-like transport equation, where the angular redistribution term in spherical geometry is treated as a source. Then, a \(P_N\) solution in plane geometry given by a combination of the solution of the associated homogeneous equation and a particular solution is developed. This is followed by a post-processing step which is very effective in improving the \(P_N\) solution. An additional amount of work with respect to that required for solving problems in plane geometry occurs in the form of a system of \(N+1\) Volterra integral equations of the second kind that has to be solved for the coefficients of the particular solution. The proposed approach has, in any case, the merit of avoiding the ill-conditioning caused by the presence of modified spherical Bessel functions in the standard \(P_N\) solution, as demonstrated by numerical results tabulated for some test cases.Fast upwind and Eulerian-Lagrangian control volume schemes for time-dependent directional space-fractional advection-dispersion equations.https://www.zbmath.org/1453.652472021-02-27T13:50:00+00:00"Du, Ning"https://www.zbmath.org/authors/?q=ai:du.ning"Guo, Xu"https://www.zbmath.org/authors/?q=ai:guo.xu"Wang, Hong"https://www.zbmath.org/authors/?q=ai:wang.hong.1Summary: We develop control volume methods for two-dimensional time-dependent advection-dominated directional space-fractional advection-dispersion equations with the directional space-fractional derivative weighted in all the directions by a probability measure in the unit circle, which are used to model the anisotropic superdiffusive transport of solutes in groundwater moving through subsurface heterogeneous porous media.
We develop a fast upwind control volume method for the governing equation to eliminate the spurious numerical oscillations that often occur in space-centered numerical discretizations of advection term, which are relatively straightforward to implement. We also develop a Eulerian-Lagrangian control-volume method for the governing equation, which symmetrizes the governing equation by combining the time-derivative term and the advection term into a material derivative term along characteristic curves. Both methods are locally mass-conservative, which are essential in these applications.
Due to the nonlocal nature of the directional space-fractional differential operators, corresponding numerical discretizations usually generate full stiffness matrices. Conventional direct solvers tend to require \(O(N^2)\) memory requirement and have \(O(N^3)\) computational complexity per time step, where \(N\) is the number of spatial unknowns, which is computationally significantly more expensive than the numerical approximations of integer-order advection-diffusion equations. Based on the analysis of the structure of stiffness matrix, we propose a fast Krylov subspace iterative solver to accelerate the numerical approximations of both the upwind and Eulerian-Lagrangian control volume methods, which reduce computational complexity from \(O(N^3)\) by a direct solver to \(O(N\log N)\) per Krylov subspace iteration per time step and a memory requirement from \(O(N^2)\) to \(O(N)\). Numerical results are presented to show the utility of the methods.A Hellinger distance approach to MCMC diagnostics.https://www.zbmath.org/1453.620502021-02-27T13:50:00+00:00"Boone, Edward L."https://www.zbmath.org/authors/?q=ai:boone.edward-l"Merrick, Jason R. W."https://www.zbmath.org/authors/?q=ai:merrick.jason-r-w"Krachey, Matthew J."https://www.zbmath.org/authors/?q=ai:krachey.matthew-jSummary: Bayesian analysis often requires the researcher to employ Markov Chain Monte Carlo (MCMC) techniques to draw samples from a posterior distribution which in turn is used to make inferences. Currently, several approaches to determine convergence of the chain as well as sensitivities of the resulting inferences have been developed. This work develops a Hellinger distance approach to MCMC diagnostics. An approximation to the Hellinger distance between two distributions \(f\) and \(g\) based on sampling is introduced. This approximation is studied via simulation to determine the accuracy. A criterion for using this Hellinger distance for determining chain convergence is proposed as well as a criterion for sensitivity studies. These criteria are illustrated using a dataset concerning the \textit{Anguilla australis}, an eel native to New Zealand.A CDG-FE method for the two-dimensional Green-Naghdi model with the enhanced dispersive property.https://www.zbmath.org/1453.760772021-02-27T13:50:00+00:00"Li, Maojun"https://www.zbmath.org/authors/?q=ai:li.maojun"Xu, Liwei"https://www.zbmath.org/authors/?q=ai:xu.liwei"Cheng, Yongping"https://www.zbmath.org/authors/?q=ai:cheng.yongpingSummary: In this work, we investigate numerical solutions of the two-dimensional shallow water wave using a fully nonlinear Green-Naghdi model with an improved dispersive effect. For numerics, the Green-Naghdi model is rewritten into a formulation coupling a pseudo-conservative system and a set of pseudo-elliptic equations. Since the pseudo-conservative system is no longer hyperbolic and its Riemann problem can only be approximately solved, we consider the utilization of the central discontinuous Galerkin method, which possesses an important feature of the needlessness of Riemann solvers. Meanwhile, the stationary elliptic part will be solved using the finite element method. Both the well-balanced and the positivity-preserving features, which are highly desirable in the simulation of the shallow water wave, will be embedded into the proposed numerical scheme. The accuracy and efficiency of the numerical model and method will be illustrated through numerical tests.A weakly nonlinear, energy stable scheme for the strongly anisotropic Cahn-Hilliard equation and its convergence analysis.https://www.zbmath.org/1453.652682021-02-27T13:50:00+00:00"Cheng, Kelong"https://www.zbmath.org/authors/?q=ai:cheng.kelong"Wang, Cheng"https://www.zbmath.org/authors/?q=ai:wang.cheng.1"Wise, Steven M."https://www.zbmath.org/authors/?q=ai:wise.steven-mSummary: In this paper we propose and analyze a weakly nonlinear, energy stable numerical scheme for the strongly anisotropic Cahn-Hilliard model. In particular, a highly nonlinear and singular anisotropic surface energy makes the PDE system very challenging at both the analytical and numerical levels. To overcome this well-known difficulty, we perform a convexity analysis on the anisotropic interfacial energy, and a careful estimate reveals that all its second order functional derivatives stay uniformly bounded by a global constant. This subtle fact enables one to derive an energy stable numerical scheme. Moreover, a linear approximation becomes available for the surface energy part, and a detailed estimate demonstrates the corresponding energy stability. Its combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear, energy stable scheme for the whole system. In particular, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, and no auxiliary variable needs to be introduced. This has important implications, for example, in the case that the method needs to satisfy a maximum principle. More importantly, with a careful application of the global bound for the second order functional derivatives, an optimal rate convergence analysis becomes available for the proposed numerical scheme, which is the first such result in this area. Meanwhile, for a Cahn-Hilliard system with a sufficiently large degree of anisotropy, a Willmore or biharmonic regularization has to be introduced to make the equation well-posed. For such a physical model, all the presented analyses are still available; the unique solvability, energy stability and convergence estimate can be derived in an appropriate manner. In addition, the Fourier pseudo-spectral spatial approximation is applied, and all the theoretical results could be extended for the fully discrete scheme. Finally, a few numerical results are presented, which confirm the robustness and accuracy of the proposed scheme.A multi-rate iterative coupling scheme for simulating dynamic ruptures and seismic waves generation in the prestressed earth.https://www.zbmath.org/1453.860302021-02-27T13:50:00+00:00"Ye, Ruichao"https://www.zbmath.org/authors/?q=ai:ye.ruichao"Kumar, Kundan"https://www.zbmath.org/authors/?q=ai:kumar.kundan"de Hoop, Maarten V."https://www.zbmath.org/authors/?q=ai:de-hoop.maarten-v"Campillo, Michel"https://www.zbmath.org/authors/?q=ai:campillo.michelSummary: We present a novel method to simulate the dynamic evolution of spontaneous ruptures governed by rate- and state-dependent friction laws and the interaction with seismic waves in a prestressed elastically deforming body. We propose a multi-rate iterative coupling scheme based on the variational form of the elastic-gravitational equations, and discretize employing a discontinuous Galerkin method, with nonlinear interior boundary conditions being weakly imposed across the fault surface as numerical fluxes. We introduce necessary interface jump penalty terms as well as an artificial viscous regularization, with the conditions for penalty and viscosity coefficients given based on an energy estimate and a convergence analysis. In the multi-rate scheme, an implicit-explicit Euler scheme in time is invoked, and the time step for the evolution of friction is chosen significantly finer than that for wave propagation and scattering. This is facilitated by the iterative scheme through the underlying decoupling where the linear, elastic wave equation plays the role of a Schur-complement to the friction model. A nonlinearly constrained optimization problem localized to each element on the rupture surface is then formulated and solved using the Gauss-Newton method. We test our algorithm on several benchmark examples and illustrate the generality of our method for realistic rupture simulations.Solving nonlinear system of second-order boundary value problems using a newly constructed scaling function.https://www.zbmath.org/1453.651942021-02-27T13:50:00+00:00"Liu, Yanan"https://www.zbmath.org/authors/?q=ai:liu.yananSummary: In this paper, a scaling function constructed by special filter coefficients is used for solving nonlinear system of second-order boundary value problems. The basis functions in interval originated from the newly constructed scaling function are directly used for function approximation. The Galerkin method and iteration approach are used for solution. Some numerical examples are presented to demonstrate the validity of the numerical technique. Numerical results prove that the new basis functions have good approximation ability and the present method is very efficient and highly accurate in solving nonlinear system of second-order boundary value problems.A new conservative finite-difference scheme for anisotropic elliptic problems in bounded domain.https://www.zbmath.org/1453.653852021-02-27T13:50:00+00:00"Soler, J. A."https://www.zbmath.org/authors/?q=ai:soler.j-a"Schwander, F."https://www.zbmath.org/authors/?q=ai:schwander.f"Giorgiani, G."https://www.zbmath.org/authors/?q=ai:giorgiani.giorgio"Liandrat, J."https://www.zbmath.org/authors/?q=ai:liandrat.jacques"Tamain, P."https://www.zbmath.org/authors/?q=ai:tamain.p"Serre, E."https://www.zbmath.org/authors/?q=ai:serre.ericSummary: Highly anisotropic elliptic problems occur in many physical models that need to be solved numerically. A direction of dominant diffusion is thus introduced (called here parallel direction) along which the diffusion coefficient is several orders larger of magnitude than in the perpendicular one. In this case, finite-difference methods based on misaligned stencils are generally not designed to provide an optimal discretization, and may lead the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion.
This paper proposes an original scheme using non-aligned Cartesian grids and interpolations aligned along a parallel diffusion direction. Here, this direction is assumed to be supported by a divergence-free vector field which never vanishes and it is supposed to be stationary in time. Based on the Support Operator Method (SOM), the self-adjointness property of the parallel diffusion operator is maintained on the discrete level. Compared with existing methods, the present formulation further guarantees the conservativity of the fluxes in both parallel and perpendicular directions. In addition, when the flow intercepts a boundary in the parallel direction, an accurate discretization of the boundary condition is presented that avoids the uncertainties of extrapolated far ghost points classically used and ensures a better accuracy of the solution. Numerical tests based on manufactured solutions show the method is able to provide accurate and stable numerical approximations in both periodic and bounded domains with a drastically reduced number of degrees of freedom with respect to non-aligned approaches.How to obtain an accurate gradient for interface problems?https://www.zbmath.org/1453.653872021-02-27T13:50:00+00:00"Tong, Fenghua"https://www.zbmath.org/authors/?q=ai:tong.fenghua"Wang, Weilong"https://www.zbmath.org/authors/?q=ai:wang.weilong"Feng, Xinlong"https://www.zbmath.org/authors/?q=ai:feng.xinlong"Zhao, Jianping"https://www.zbmath.org/authors/?q=ai:zhao.jianping"Li, Zhilin"https://www.zbmath.org/authors/?q=ai:li.zhilinSummary: It is well-known that the Immersed Interface Method (IIM) is second order accurate for interface problems. But the accuracy of the first order derivatives, or gradients for short, is not so clear and is often assumed to be first order accurate. In this paper, new strategies based on IIM are proposed for elliptic interface problems to compute the gradient at grid points both regular and irregular, and at the interface from each side of the interface. Second order in 1D, or nearly second order (except a factor of \(|\log h|)\) convergence in 2D of the computed gradient is obtained with almost no extra cost, and has been explained in intuition and verified by non-trivial numerical tests. Numerical examples in one, two dimensions, radial and axis-symmetric cases in polar and spherical coordinates are presented to validate the numerical methods and analysis.A study on spectral methods for linear and nonlinear fractional differential equations.https://www.zbmath.org/1453.651892021-02-27T13:50:00+00:00"Behroozifar, Mahmoud"https://www.zbmath.org/authors/?q=ai:behroozifar.mahmoud"Ahmadpour, Farkhondeh"https://www.zbmath.org/authors/?q=ai:ahmadpour.farkhondehSummary: In this paper, a computational method based on the spectral methods with shifted Jacobi polynomials is applied for the numerical solution of the linear and nonlinear multi-order fractional differential equations. Fractional derivative is described in the Caputo sense. Operational matrix of fractional differential of shifted Jacobi polynomials is stated. This matrix together with the tau method and collocation method are utilised to reduce the linear and nonlinear fractional differential equations to a system of algebraic equations, respectively. The purpose of this paper is to make a comparison between this simple method and other existing methods to show the performance and preciseness of the presented method. Due to this, we used this technique for some illustrative numerical tests which the results demonstrate the validity and efficiency of the method.A minimal stabilization procedure for isogeometric methods on trimmed geometries.https://www.zbmath.org/1453.654032021-02-27T13:50:00+00:00"Buffa, A."https://www.zbmath.org/authors/?q=ai:buffa.annalisa|buffa.andrea-m"Puppi, R."https://www.zbmath.org/authors/?q=ai:puppi.r"Vázquez, R."https://www.zbmath.org/authors/?q=ai:vazquez.rafael|vazquez.rogelio|vazquez.roberto|vazquez.rosendo|vazquez.r-rodriguez|vazquez.r-aThe authors are firstly concerned with an isogeometric formulation of a mixed (Dirichlet and Neumann) problem attached to the Laplacian. Then, they explain the main challenges one need to face, namely, integration, conditioning, and numerical stability of the associated linear system when the above model is solved by FEM on a trimmed domain. As the main ingredient, they introduce a new stabilization technique which consists in locally modifying the weak formulation while keeping the discrete functional space unaffected. Two different versions of the stabilization are introduced. The first one is based on polynomial extrapolation in the parametric domain and the second one is a projection-based stabilization, performed directly on the physical domain. The latter allows an optimal recover of the priori error estimates. Some numerical experiments are carried out using the MATLAB library GeoPDEs. They validate the effectiveness of the introduced stabilization technique.
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)A parameter-uniform first order convergent numerical method for a semi-linear system of singularly perturbed second order delay differential equations.https://www.zbmath.org/1453.651472021-02-27T13:50:00+00:00"Manikandan, Mariappan"https://www.zbmath.org/authors/?q=ai:manikandan.mariappan"Miller, John J. H."https://www.zbmath.org/authors/?q=ai:miller.john-j-h"Sigamani, Valarmathi"https://www.zbmath.org/authors/?q=ai:sigamani.valarmathiSummary: In this paper, a boundary value problem for a semi-linear system of two singularly perturbed second order delay differential equations is considered on the interval \((0, 2)\). The components of the solution of this system exhibit boundary layers at \(x=0\) and \(x=2\) and interior layers at \(x=1\). A numerical method composed of a classical finite difference operator applied on a piecewise uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical computation is described, which supports the theoretical results.
For the entire collection see [Zbl 1354.65008].A parameter uniform numerical method for an initial value problem for a system of singularly perturbed delay differential equations with discontinuous source terms.https://www.zbmath.org/1453.651492021-02-27T13:50:00+00:00"Shivaranjani, Nagarajan"https://www.zbmath.org/authors/?q=ai:shivaranjani.nagarajan"Miller, John J. H."https://www.zbmath.org/authors/?q=ai:miller.john-j-h"Sigamani, Valarmathi"https://www.zbmath.org/authors/?q=ai:sigamani.valarmathiSummary: In this paper an initial value problem for a system of singularly perturbed first order delay differential equations with discontinuous source terms is considered on the interval \((0, 2]\). The source terms are assumed to have simple discontinuities at the point \(d \in (0, 2)\). The components of the solution exhibit initial layers and interior layers. The interior layers occuring in the solution are of two types-interior layers due to delay and interior layers due to the discontinuity of the source terms. A numerical method composed of the standard backward difference operator and a piecewise-uniform Shishkin mesh which resolves the initial and interior layers is suggested. This method is proved to be essentially first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical illustrations are provided to support the theory.
For the entire collection see [Zbl 1354.65008].Numerical method for a singularly perturbed boundary value problem for a linear parabolic second order delay differential equation.https://www.zbmath.org/1453.651842021-02-27T13:50:00+00:00"Swaminathan, Parthiban"https://www.zbmath.org/authors/?q=ai:swaminathan.parthiban"Sigamani, Valarmathi"https://www.zbmath.org/authors/?q=ai:sigamani.valarmathi"Victor, Franklin"https://www.zbmath.org/authors/?q=ai:victor.franklinSummary: A singularly perturbed boundary value problem for a linear parabolic second order delay differential equation of reaction-diffusion type is considered. As the highest order space derivative is multiplied by a singular perturbation parameter, the solution exhibits boundary layers. Also, the delay term that occurs in the space variable gives rise to interior layers. A numerical method which uses classical finite difference scheme on a Shishkin piecewise uniform mesh is suggested to approximate the solution. The method is proved to be first order convergent uniformly with respect to the singular perturbation parameter. Numerical illustrations are also presented.
For the entire collection see [Zbl 1354.65008].A numerical method for a system of singularly perturbed differential equations of reaction-diffusion type with negative shift.https://www.zbmath.org/1453.651812021-02-27T13:50:00+00:00"Selvi, P. Avudai"https://www.zbmath.org/authors/?q=ai:avudai-selvi.p"Narasimhan, Ramanujam"https://www.zbmath.org/authors/?q=ai:narasimhan.ramanujamSummary: A numerical method based on an iterative scheme is proposed for a system of singularly perturbed differential equations of reaction-diffusion type with negative shift term. In this method the solution of the delay problem is obtained as the limit of the solutions to a sequence of the non-delay problems. Then non-delay problems are solved by applying available finite difference scheme and finite element method in the literature. An error estimate in supremum norm is derived. Numerical experiments are carried out.
For the entire collection see [Zbl 1354.65008].A local conservative multiscale method for elliptic problems with oscillating coefficients.https://www.zbmath.org/1453.654092021-02-27T13:50:00+00:00"Jeon, Youngmok"https://www.zbmath.org/authors/?q=ai:jeon.youngmok"Park, Eun-Jae"https://www.zbmath.org/authors/?q=ai:park.eun-jaeSummary: A new multiscale finite element method for elliptic problems with highly oscillating coefficients are introduced. A hybridization yields a locally flux-conserving numerical scheme for multiscale problems. Our approach naturally induces a homogenized equation which facilitates error analysis. Complete convergence analysis is given and numerical examples are presented to validate our analysis.Initial or boundary value problems for systems of singularly perturbed differential equations and their solution profile.https://www.zbmath.org/1453.651832021-02-27T13:50:00+00:00"Sigamani, Valarmathi"https://www.zbmath.org/authors/?q=ai:sigamani.valarmathiSummary: Singular perturbation problems, by nature, are not easy to handle and they demand efficient techniques to solve and careful analysis. And systems of singular perturbation problems are tougher as their solutions exhibit layers with sub-layers. Their properties are studied and examples are given to illustrate.
For the entire collection see [Zbl 1354.65008].Estimating the diagonal of matrix functions.https://www.zbmath.org/1453.650922021-02-27T13:50:00+00:00"Fika, Paraskevi"https://www.zbmath.org/authors/?q=ai:fika.paraskevi"Mitrouli, Marilena"https://www.zbmath.org/authors/?q=ai:mitrouli.marilena"Roupa, Paraskevi"https://www.zbmath.org/authors/?q=ai:roupa.paraskeviSummary: The evaluation of the diagonal of matrix functions arises in many applications and an efficient approximation of it, without estimating the whole matrix \(f(A)\), would be useful. In the present paper, we compare and analyze the performance of three numerical methods adjusted to attain the estimation of the diagonal of matrix functions \(f(A)\), where \(A\in\mathbb{R}^{p\times p}\) is a symmetric matrix and \(f\) a suitable function. The applied numerical methods are based on extrapolation and Gaussian quadrature rules. Various numerical results illustrating the effectiveness of these methods and insightful remarks about their complexity and accuracy are demonstrated.Interior layers in singularly perturbed problems.https://www.zbmath.org/1453.651782021-02-27T13:50:00+00:00"O'Riordan, Eugene"https://www.zbmath.org/authors/?q=ai:oriordan.eugeneSummary: To construct layer adapted meshes for a class of singularly perturbed problems, whose solutions contain boundary layers, it is necessary to identify both the location and the width of any boundary layers present in the solution. Additional interior layers can appear when the data for the problem is not sufficiently smooth. In the context of singularly perturbed partial differential equations, the presence of any interior layer typically requires the introduction of a transformation of the problem, which facilitates the necessary alignment of the mesh to the trajectory of the interior layer. Here we review a selection of published results on such problems to illustrate the variety of ways that interior layers can appear.
For the entire collection see [Zbl 1354.65008].Elementary tutorial on numerical methods for singular perturbation problems.https://www.zbmath.org/1453.651772021-02-27T13:50:00+00:00"Miller, John J. H."https://www.zbmath.org/authors/?q=ai:miller.john-j-hSummary: In the first section we introduce a simple singularly perturbed initial value problem for a first order linear differential equation. We construct the backward Euler finite difference method for this problem. We then discuss continuous and discrete maximum principles for the associated continuous and discrete operators and we conclude the section by defining what is meant by a parameter-uniform numerical method. In the second section we introduce a fitted operator method on a uniform mesh for our simple initial value problem defined in the previous section. We then prove rigorously that this method is parameter-uniform at the mesh points. Fitted mesh methods on piecewise uniform meshes are introduced in the third section. A fitted mesh method for our simple initial value problem is constructed. It is proved rigorously that this method is parameter-uniform at the mesh points. Finally, in the fourth section, numerical solutions of singular perturbation problems are discussed. Computations using standard and a parameter-uniform numerical method are presented. The usefulness and reliability of parameter-uniform methods is demonstrated.
For the entire collection see [Zbl 1354.65008].Quadratic numerical treatment for singular integral equations with logarithmic kernel.https://www.zbmath.org/1453.654582021-02-27T13:50:00+00:00"Nadir, Mostefa"https://www.zbmath.org/authors/?q=ai:nadir.mostefa"Gagui, Bachir"https://www.zbmath.org/authors/?q=ai:gagui.bachirSummary: The goal of this paper is to present a direct method for an approximative solution of a weakly singular integral equations (WSIE) with logarithmic kernel on a piecewise smooth integration path using a modified quadratic spline approximation, we also show that this approximation gives an efficient approach to the analytical solution of WSIE.Fourth order computational method for two parameters singularly perturbed boundary value problem using non-polynomial cubic spline.https://www.zbmath.org/1453.651792021-02-27T13:50:00+00:00"Phaneendra, K."https://www.zbmath.org/authors/?q=ai:phaneendra.kolloju"Mahesh, G."https://www.zbmath.org/authors/?q=ai:mahesh.gSummary: In this paper, we proposed a fourth order finite difference scheme using non-polynomial cubic spline for the solution of two parameters singularly perturbed two-point boundary value problem having dual boundary layer on a uniform mesh. In this method, the first order derivatives in the non-polynomial cubic spline finite difference scheme are replaced by the higher order finite differences to get the discretisation equation for the problem. The discretisation equation is solved by the tridiagonal solver discrete invariant imbedding. The proposed method is analysed for convergence and a fourth order rate of convergence is proved. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods.Optimal error estimate of a decoupled conservative local discontinuous Galerkin method for the Klein-Gordon-Schrödinger equations.https://www.zbmath.org/1453.652832021-02-27T13:50:00+00:00"Yang, He"https://www.zbmath.org/authors/?q=ai:yang.heSummary: In this paper, we propose a decoupled local discontinuous Galerkin method for solving the Klein-Gordon-Schrödinger (KGS) equations. The KGS equations is a model of the Yukawa interaction of complex scalar nucleons and real scalar mesons. The advantage of our scheme is that the computation of the nucleon and meson field is fully decoupled, so that it is especially suitable for parallel computing. We present the conservation property of our fully discrete scheme, including the energy and Hamiltonian conservation, and establish the optimal error estimate.Analytical solutions to nonlinear problems by the generalised form of HAM: a note.https://www.zbmath.org/1453.652052021-02-27T13:50:00+00:00"Shukla, Anant Kant"https://www.zbmath.org/authors/?q=ai:shukla.anant-kant"Ramamohan, T. R."https://www.zbmath.org/authors/?q=ai:ramamohan.t-r"Srinivas, S."https://www.zbmath.org/authors/?q=ai:srinivas.sampath|srinivas.suripeddi|srinivas.sampalli|srinivas.s-t-p-t|srinivas.sindhu|srinivas.s-sSummary: The objective of this article is to obtain analytical solutions for a set of nonlinear problems by using `further generalisation of HAM'. In comparison to the Homotopy analysis method (HAM) solutions, more accurate solutions are obtained by introducing an extra term in the frame of HAM. We consider a set of three nonlinear problems of which first two are governed by single nonlinear ordinary differential equation (they are two cases of the forced Van der Pol Duffing oscillator) and third one is governed by a system of four coupled nonlinear ordinary differential equations. A maximum reduction of approximately 25\% in the square residual error is obtained by using the generalised form of HAM compared to the square residual error without the generalised form.On the reflection of solitons of the cubic nonlinear Schrödinger equation.https://www.zbmath.org/1453.653292021-02-27T13:50:00+00:00"Katsaounis, Theodoros"https://www.zbmath.org/authors/?q=ai:katsaounis.theodoros"Mitsotakis, Dimitrios"https://www.zbmath.org/authors/?q=ai:mitsotakis.dimitrios-eSummary: In this paper, we perform a numerical study on the interesting phenomenon of soliton reflection of solid walls. We consider the 2D cubic nonlinear Schrödinger equation as the underlying mathematical model, and we use an implicit-explicit type Crank-Nicolson finite element scheme for its numerical solution. After verifying the perfect reflection of the solitons on a vertical wall, we present the imperfect reflection of a dark soliton on a diagonal wall.Self-localized solitons of a \(q\)-deformed quantum system.https://www.zbmath.org/1453.351602021-02-27T13:50:00+00:00"Bayındır, Cihan"https://www.zbmath.org/authors/?q=ai:bayindir.cihan"Altintas, Azmi Ali"https://www.zbmath.org/authors/?q=ai:altintas.azmi-ali"Ozaydin, Fatih"https://www.zbmath.org/authors/?q=ai:ozaydin.fatihSummary: Beyond a pure mathematical interest, \(q\)-deformation is promising for the modeling and interpretation of various physical phenomena. In this paper, we numerically investigate the existence and properties of the self-localized soliton solutions of the nonlinear Schrödinger equation (NLSE) with a \(q\)-deformed Rosen-Morse potential. By implementing a Petviashvili method (PM), we obtain the self-localized one and two soliton solutions of the NLSE with a \(q\)-deformed Rosen-Morse potential. In order to investigate the temporal behavior and stabilities of these solitons, we implement a Fourier spectral method with a 4th order Runge-Kutta time integrator. We observe that the self-localized one and two solitons are stable and remain bounded with a pulsating behavior and minor changes in the sidelobes of the soliton waveform. Additionally, we investigate the stability and robustness of these solitons under noisy perturbations. A sinusoidal monochromatic wave field modeled within the frame of the NLSE with a \(q\)-deformed Rosen-Morse potential turns into a chaotic wavefield and exhibits rogue oscillations due to modulation instability triggered by noise, however, the self-localized solitons of the NLSE with a \(q\)-deformed Rosen-Morse potential are stable and robust under the effect of noise. We also show that soliton profiles can be reconstructed after a denoising process performed using a Savitzky-Golay filter.A spline-based computational technique applicable for solution of boundary value problem arising in human physiology.https://www.zbmath.org/1453.920072021-02-27T13:50:00+00:00"Srivastava, Pankaj Kumar"https://www.zbmath.org/authors/?q=ai:srivastava.pankaj-kumarSummary: Non-polynomial quintic spline functions based algorithms are used for computing an approximation to the nonlinear two point second order singular boundary value problems arising in human physiology. After removing the singularity by L'Hospital rule, the resulting boundary value problem is then efficiently treated by employing non-polynomial quintic spline for finding the numerical solution. Two examples have been included and comparison of the numerical results made with cubic extended B-spline method and finite difference method.Numerical solution of fuzzy differential equations using orthogonal polynomials.https://www.zbmath.org/1453.651972021-02-27T13:50:00+00:00"Tapaswini, Smita"https://www.zbmath.org/authors/?q=ai:tapaswini.smita"Chakraverty, Snehashish"https://www.zbmath.org/authors/?q=ai:chakraverty.snehashishSummary: The present paper proposed a new method to solve \(n\)th order fuzzy differential equations using collocation type of method. In the solution procedure, Gram Schmidt orthogonalisation process is used with Legendre and Chebyshev polynomials. Known example problems are solved and compared with the exact results to illustrate the efficiency and reliability of the proposed method.Resolution of nonlinear and non-autonomous ODEs by the ADM using a new practical Adomian polynomials.https://www.zbmath.org/1453.652062021-02-27T13:50:00+00:00"Zaouagui, Noureddine"https://www.zbmath.org/authors/?q=ai:zaouagui.noureddine"Badredine, Toufik"https://www.zbmath.org/authors/?q=ai:badredine.toufikSummary: In this paper, a new practical formulas of Adomian polynomials has been adapted to resolve nonlinear and non-autonomous ordinary differential equations by the Adomian decomposition method, a simple computational for this new polynomials has been suggested, Therefore new conditions of convergence have been generalized.New cascadic multigrid methods for two-dimensional Poisson problem based on the fourth-order compact difference scheme.https://www.zbmath.org/1453.653832021-02-27T13:50:00+00:00"Li, Ming"https://www.zbmath.org/authors/?q=ai:li.ming.6"Li, Chenliang"https://www.zbmath.org/authors/?q=ai:li.chenliangSummary: Based on the fourth-order compact finite difference scheme, new extrapolation cascadic multigrid methods for two-dimensional Poisson problem are presented. In these new methods, a new extrapolation operator and a spline interpolation operator are used to provide a better initial value on refined grid. Numerical experiments show the new methods have higher accuracy and better efficiency.A unified a posteriori error estimator for finite volume methods for the Stokes equations.https://www.zbmath.org/1453.653922021-02-27T13:50:00+00:00"Wang, Junping"https://www.zbmath.org/authors/?q=ai:wang.junping"Wang, Yanqiu"https://www.zbmath.org/authors/?q=ai:wang.yanqiu"Ye, Xiu"https://www.zbmath.org/authors/?q=ai:ye.xiuSummary: In this paper, the authors established a unified framework for deriving and analyzing a posteriori error estimators for finite volume methods for the Stokes equations. The a posteriori error estimators are residual based and are applicable to various finite volume methods for the Stokes equations. In particular, the unified theoretical analysis works well for finite volume schemes arising from using trial functions of conforming, nonconforming, and discontinuous finite element functions, yielding new results that are not seen in the existing literature.Solving a Bernoulli type free boundary problem with random diffusion.https://www.zbmath.org/1453.351992021-02-27T13:50:00+00:00"Brügger, Rahel"https://www.zbmath.org/authors/?q=ai:brugger.rahel"Croce, Roberto"https://www.zbmath.org/authors/?q=ai:croce.roberto"Harbrecht, Helmut"https://www.zbmath.org/authors/?q=ai:harbrecht.helmutThe paper deals with a Bernoulli's free boundary problem under modelling uncertainties. The forward problem is governed by an overdetermined diffusive equation. The basic idea consists in rewriting it as a shape optimization problem. In particular, a shape functional measuring the expected energy is minimized with respect to the shape of the domain. The associated shape gradient is derived by using the standard velocity method. The resulting sensitivity is given by quantities concentrated on the moving boundary depending on the solution to the state equation, which fits the so-called Hadamard's Structure Theorem. In addition, random diffusion is considered, which may represent modelling uncertainties, for instance. The resulting high-dimensional integral is approximated by the quasi-Monte Carlo method. A level-set method driven by the obtained shape derivative is proposed. Finally, a nice set of numerical experiments is presented, showing that uncertainties in the diffusive coefficient may lead to significantly different minimizers, as expected.
Reviewer: Antonio André Novotny (Petrópolis)Random set modelling of three-dimensional objects in a hierarchical Bayesian context.https://www.zbmath.org/1453.621572021-02-27T13:50:00+00:00"Micheas, Athanasios C."https://www.zbmath.org/authors/?q=ai:micheas.athanasios-christou"Wikle, Christopher K."https://www.zbmath.org/authors/?q=ai:wikle.christopher-k"Larsen, David R."https://www.zbmath.org/authors/?q=ai:larsen.david-rSummary: We present a Bayesian approach to modelling and estimating objects in three dimensions. A general stochastic model for object recognition based on points in the domain of observation is created via hierarchical mixtures, allowing for the inclusion of important prior information about the objects under consideration. Objects under consideration are created based on three-dimensional (3D) points in space, with each point having some probability of membership to different objects. This is the first object-oriented statistical approach that models 3D objects in a true 3D environment. A data-augmentation approach and a Birth-Death Markov Chain Monte Carlo algorithm are incorporated to provide classification probabilities of each data point to one or more of the identified objects and to obtain estimates of the parameters that describe each object. Strengths of the methodology include its ease in accommodating different data and process models, its flexibility in handling varying numbers of mixture components, and most importantly, its ability to allow for inference on individual characteristics of objects. These strengths are demonstrated on an application for a forest (tree objects) near Olympia, WA, based on remotely sensed Light Detection and Ranging point observations.A correction function method for Poisson problems with interface jump conditions.https://www.zbmath.org/1453.350542021-02-27T13:50:00+00:00"Marques, Alexandre Noll"https://www.zbmath.org/authors/?q=ai:marques.alexandre-noll"Nave, Jean-Christophe"https://www.zbmath.org/authors/?q=ai:nave.jean-christophe"Rosales, Rodolfo Ruben"https://www.zbmath.org/authors/?q=ai:rosales.rodolfo-rubenSummary: In this paper we present a method to treat interface jump conditions for constant coefficients Poisson problems that allows the use of standard ''black box'' solvers, without compromising accuracy. The basic idea of the new approach is similar to the Ghost Fluid Method (GFM). The GFM relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface. If the corrections are solution-independent, they can be moved to the right-hand-side (RHS) of the equations, producing a problem with the same linear system as if there were no jumps, only with a different RHS. However, achieving high accuracy is very hard (if not impossible) with the ''standard'' approaches used to compute the GFM correction terms.
In this paper we generalize the GFM correction terms to a correction function, defined on a band around the interface. This function is then shown to be characterized as the solution to a PDE, with appropriate boundary conditions. This PDE can, in principle, be solved to any desired order of accuracy. As an example, we apply this new method to devise a 4th order accurate scheme for the constant coefficients Poisson equation with discontinuities in 2D. This scheme is based on (i) the standard 9-point stencil discretization of the Poisson equation, (ii) a representation of the correction function in terms of bicubics, and (iii) a solution of the correction function PDE by a least squares minimization. Several applications of the method are presented to illustrate its robustness dealing with a variety of interface geometries, its capability to capture sharp discontinuities, and its high convergence rate.Conditional gradient method for double-convex fractional programming matrix problems.https://www.zbmath.org/1453.651302021-02-27T13:50:00+00:00"Bouhamidi, Abderrahman"https://www.zbmath.org/authors/?q=ai:bouhamidi.abderrahman"Bellalij, Mohammed"https://www.zbmath.org/authors/?q=ai:bellalij.mohammed"Enkhbat, Rentsen"https://www.zbmath.org/authors/?q=ai:enkhbat.rentsen"Jbilou, Khalid"https://www.zbmath.org/authors/?q=ai:jbilou.khalide"Raydan, Marcos"https://www.zbmath.org/authors/?q=ai:raydan.marcosThe authors consider the problem of optimizing the ratio of two convex functions over a closed and convex set in the space of matrices. In general, the objective function is nonconvex but, nevertheless, the problem has some special features. Taking advantage of these features, a conditional gradient method is proposed and analyzed, which is suitable for matrix problems. Numerical experiments are presented.
Reviewer: Hans Benker (Merseburg)An improved lower bound for general position subset selection.https://www.zbmath.org/1453.650472021-02-27T13:50:00+00:00"Rudi, Ali Gholami"https://www.zbmath.org/authors/?q=ai:rudi.ali-gholamiSummary: In the general position subset selection (GPSS) problem, the goal is to find the largest possible subset of a set of points, such that no three of its members are collinear. If \(s\) is the size the optimal solution, the square root of \(s\) is the current best guarantee for the size of the solution obtained using a polynomial time algorithm. In this paper, we present an algorithm for GPSS to improve this bound based on the number of collinear pairs of points. We experimentally evaluate this and few other GPSS algorithms; the result of these experiments suggests further opportunities for obtaining tighter lower bounds for GPSS.Nonstandard finite difference method revisited and application to the Ebola virus disease transmission dynamics.https://www.zbmath.org/1453.651852021-02-27T13:50:00+00:00"Anguelov, R."https://www.zbmath.org/authors/?q=ai:anguelov.roumen"Berge, T."https://www.zbmath.org/authors/?q=ai:berge.tsanou"Chapwanya, M."https://www.zbmath.org/authors/?q=ai:chapwanya.michael"Djoko, J. K."https://www.zbmath.org/authors/?q=ai:djoko.jules-k"Kama, P."https://www.zbmath.org/authors/?q=ai:kama.p"Lubuma, J. M.-S."https://www.zbmath.org/authors/?q=ai:lubuma.jean-mbaro-saman"Terefe, Y."https://www.zbmath.org/authors/?q=ai:terefe.yibeltal-adaneSummary: We provide effective and practical guidelines on the choice of the complex denominator function of the discrete derivative as well as on the choice of the nonlocal approximation of nonlinear terms in the construction of nonstandard finite difference (NSFD) schemes. Firstly, we construct nonstandard one-stage and two-stage theta methods for a general dynamical system defined by a system of autonomous ordinary differential equations. We provide a sharp condition, which captures the dynamics of the continuous model. We discuss at length how this condition is pivotal in the construction of the complex denominator function. We show that the nonstandard theta methods are elementary stable in the sense that they have exactly the same fixed-points as the continuous model and they preserve their stability, irrespective of the value of the step size. For more complex dynamical systems that are dissipative, we identify a class of nonstandard theta methods that replicate this property. We apply the first part by considering a dynamical system that models the Ebola Virus Disease (EVD). The formulation of the model involves both the fast/direct and slow/indirect transmission routes. Using the specific structure of the EVD model, we show that, apart from the guidelines in the first part, the nonlocal approximation of nonlinear terms is guided by the productive-destructive structure of the model, whereas the choice of the denominator function is based on the conservation laws and the sub-equations that are associated with the model. We construct a NSFD scheme that is dynamically consistent with respect to the properties of the continuous model such as: positivity and boundedness of solutions; local and/or global asymptotic stability of disease-free and endemic equilibrium points; dependence of the severity of the infection on self-protection measures. Throughout the paper, we provide numerical simulations that support the theory.Convergence analysis of processes with valiant projection operators in Hilbert space.https://www.zbmath.org/1453.651312021-02-27T13:50:00+00:00"Censor, Yair"https://www.zbmath.org/authors/?q=ai:censor.yair"Mansour, Rafiq"https://www.zbmath.org/authors/?q=ai:mansour.rafiqConvex feasibility problems require to find a point in the intersection of a finite family of convex sets. The authors propose to solve such problems by performing set-enlargements and apply a new kind of projection operators called valiant projectors. Furthermore, the authors study properties of valiant projectures and prove convergence of the new valiant projections method.
Reviewer: Hans Benker (Merseburg)Effective shape optimization of Laplace eigenvalue problems using domain expressions of Eulerian derivatives.https://www.zbmath.org/1453.654002021-02-27T13:50:00+00:00"Zhu, Shengfeng"https://www.zbmath.org/authors/?q=ai:zhu.shengfengThe author considers a shape optimization issue for an eigenvalue Dirichlet boundary value problem attached to the Laplace equation. For this problem, a standard Ritz-Galerkin f. e. m. approximation along with its weak variational formulation are used during shape evolution. Then the author analyses two cases, a geometry constrained one and another unconstrained. Both are solved by two shape gradient descent algorithms based on shape sensitive analysis. Some interesting numerical experiments are carried out.
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)Local convergence of a fast Steffensen-type method on Banach space under weak conditions.https://www.zbmath.org/1453.651212021-02-27T13:50:00+00:00"Argyros, Ioannis K."https://www.zbmath.org/authors/?q=ai:argyros.ioannis-konstantinos"George, Santhosh"https://www.zbmath.org/authors/?q=ai:george.santhoshSummary: This paper is devoted to the study of the seventh-order Steffensen-type methods for solving nonlinear equations in Banach spaces. Using the idea of a restricted convergence domain, we extended the applicability of the seventh-order Steffensen-type methods. Our convergence conditions are weaker than the conditions used in the earlier studies. Numerical examples are also given in this study.The variable HSS iteration based on the bacterial foraging optimisation algorithm.https://www.zbmath.org/1453.650732021-02-27T13:50:00+00:00"Meng, Guo-Yan"https://www.zbmath.org/authors/?q=ai:meng.guoyan"Zhao, Qing-Shan"https://www.zbmath.org/authors/?q=ai:zhao.qingshan"Hu, Yulan"https://www.zbmath.org/authors/?q=ai:hu.yulanSummary: In the Hermitian and skew-Hermitian splitting (HSS) iteration method, the determination of the optimal parameter is a tough task when solving a large sparse non-Hermitian positive definite linear systems. In this paper, we present the variable HSS iteration with the non-fixed positive constant. For obtaining the approximate optimal parameters of this method, we solve the minimising the optimisation model based on the bacterial foraging optimisation (BFO) algorithm. Numerical experiments have shown that the new strategy is feasible and effective than the HSS iteration method.Numerical issues in gas flow dynamics with hydraulic shocks using high order finite volume WENO schemes.https://www.zbmath.org/1453.761122021-02-27T13:50:00+00:00"Uilhoorn, F. E."https://www.zbmath.org/authors/?q=ai:uilhoorn.ferdinand-evertSummary: Accurate and efficient simulation of the hydraulic shock phenomenon in pipeline systems is of paramount importance. Even though the conservation-law formulation of the governing equations is here strongly advocated, the nonconservative form is still frequently used. This also concerns its mathematical conservative form. We investigated the numerical consequences of using the compressible gas flow model in the latter form while simulating a hydraulic shock. In this context, we also solved two Riemann problems. For the investigation, we used the third-, fifth- and seventh-order accurate weighted essentially non-oscillatory (WENO) scheme along with the Lax-Friedrichs solver at the cell interfaces. Both the classical finite volume WENO scheme and its modification WENO-Z have been implemented. A procedure based on the method of manufactured solutions has been developed to verify whether the numerical code solved correctly the hyperbolic set of equations. We demonstrated that the solutions of the conservative and nonconservative formulations are similar if we have smooth variations in the solution domain. The convective inertia term in the momentum equation should not be ignored. In the presence of shocks, differences in oscillating behavior and slope steepness near the discontinuities were observed. For the hydraulic shock problem, spurious oscillations appeared while using the nonconservative formulation in combination with the WENO-Z reconstruction.Adaptive activation functions accelerate convergence in deep and physics-informed neural networks.https://www.zbmath.org/1453.681652021-02-27T13:50:00+00:00"Jagtap, Ameya D."https://www.zbmath.org/authors/?q=ai:jagtap.ameya-d"Kawaguchi, Kenji"https://www.zbmath.org/authors/?q=ai:kawaguchi.kenji"Karniadakis, George Em"https://www.zbmath.org/authors/?q=ai:karniadakis.george-emSummary: We employ adaptive activation functions for regression in deep and physics-informed neural networks (PINNs) to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations. In particular, we solve the nonlinear Klein-Gordon equation, which has smooth solutions, the nonlinear Burgers equation, which can admit high gradient solutions, and the Helmholtz equation. We introduce a scalable hyper-parameter in the activation function, which can be optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process. The adaptive activation function has better learning capabilities than the traditional one (fixed activation) as it improves greatly the convergence rate, especially at early training, as well as the solution accuracy. To better understand the learning process, we plot the neural network solution in the frequency domain to examine how the network captures successively different frequency bands present in the solution. We consider both forward problems, where the approximate solutions are obtained, as well as inverse problems, where parameters involved in the governing equation are identified. Our simulation results show that the proposed method is a very simple and effective approach to increase the efficiency, robustness and accuracy of the neural network approximation of nonlinear functions as well as solutions of partial differential equations, especially for forward problems. We theoretically prove that in the proposed method, gradient descent algorithms are not attracted to suboptimal critical points or local minima. Furthermore, the proposed adaptive activation functions are shown to accelerate the minimization process of the loss values in standard deep learning benchmarks using CIFAR-10, CIFAR-100, SVHN, MNIST, KMNIST, Fashion-MNIST, and Semeion datasets with and without data augmentation.Numerical solution of the Schrödinger equations via a reliable algorithm.https://www.zbmath.org/1453.652182021-02-27T13:50:00+00:00"Heidari, Mohammad"https://www.zbmath.org/authors/?q=ai:heidari.mohammad"Borzabadi, Akbar Hashemi"https://www.zbmath.org/authors/?q=ai:borzabadi.akbar-hashemiSummary: In this paper, a reliable algorithm for solving Schrödinger equations is established. By second-order central difference scheme, the second-order spatial partial derivative of the Schrödinger equations are reduced to a system of first-order ordinary differential equations, that are solved by an efficient algorithm. The comparison of the numerical solution and the exact solution for some test cases shows that the given algorithm is easy and practical for extracting good approximate solutions of Schrödinger equations.Conservative multirate multiscale simulation of multiphase flow in heterogeneous porous media.https://www.zbmath.org/1453.760912021-02-27T13:50:00+00:00"Delpopolo Carciopolo, Ludovica"https://www.zbmath.org/authors/?q=ai:delpopolo-carciopolo.ludovica"Formaggia, Luca"https://www.zbmath.org/authors/?q=ai:formaggia.luca"Scotti, Anna"https://www.zbmath.org/authors/?q=ai:scotti.anna"Hajibeygi, Hadi"https://www.zbmath.org/authors/?q=ai:hajibeygi.hadiSummary: Accurate and efficient simulation of multiphase flow in heterogeneous porous media motivates the development of space-time multiscale strategies for the coupled nonlinear flow (pressure) and saturation transport equations. The flow equation entails heterogeneous high-resolution (fine-scale) coefficients and is global (elliptic or parabolic). The time-dependent saturation profile, on the other hand, may exhibit sharp local gradients or discontinuities (fronts) where the solution accuracy is highly sensitive to the time-step size. Therefore, accurate flow solvers need to address the multiscale spatial scales, while advanced transport solvers need to also tackle multiple time scales. This paper presents the first multirate multiscale method for space-time conservative multiscale simulation of sequentially coupled flow and transport equations. The method computes the pressure equation at the coarse spatial scale with a multiscale finite volume technique, while the transport equation is solved by taking variable time-step sizes at different locations of the domain. At each coarse time step, the developed local time-stepping technique employs an adaptive recursive time step refinement to capture the fronts accurately. The applicability (accuracy and efficiency) of the method is investigated for a wide range of two-phase flow simulations in heterogeneous porous media. For the studied cases, the proposed method is found to provide 3 to 4 times faster simulations. Therefore, it provides a promising strategy to minimize the tradeoff between accuracy and efficiency for field-scale applications.Generalized multiscale approximation of mixed finite elements with velocity elimination for subsurface flow.https://www.zbmath.org/1453.760652021-02-27T13:50:00+00:00"Chen, Jie"https://www.zbmath.org/authors/?q=ai:chen.jie.6|chen.jie|chen.jie.10|chen.jie.2|chen.jie.7|chen.jie.9|chen.jie.8|chen.jie.5|chen.jie.4|chen.jie.3|chen.jie.1"Chung, Eric T."https://www.zbmath.org/authors/?q=ai:chung.eric-t"He, Zhengkang"https://www.zbmath.org/authors/?q=ai:he.zhengkang"Sun, Shuyu"https://www.zbmath.org/authors/?q=ai:sun.shuyuSummary: A frame work of the mixed generalized multiscale finite element method (GMsFEM) for solving Darcy's law in heterogeneous media is studied in this paper. Our approach approximates pressure in multiscale function space that is between fine-grid space and coarse-grid space and solves velocity directly in the fine-grid space. To construct multiscale basis functions for each coarse-grid element, three types of snapshot space are raised. The first one is taken as the fine-grid space for pressure and the other two cases need to solve a local problem on each coarse-grid element. We describe a spectral decomposition in the snapshot space motivated by the analysis to further reduce the dimension of the space that is used to approximate the pressure. Since the velocity is directly solved in the fine-grid space, in the linear system for the mixed finite elements, the velocity matrix can be approximated by a diagonal matrix without losing any accuracy. Thus it can be inverted easily. This reduces computational cost greatly and makes our scheme simple and easy for application. Comparing to our previous work of mixed generalized multiscale finite element method [the second author et al., Multiscale Model. Simul. 13, No. 1, 338--366 (2015; Zbl 1317.65204)], both the pressure and velocity space in this approach are bigger. As a consequence, this method enjoys better accuracy. While the computational cost does not increase because of the good property of velocity matrix. Moreover, the proposed method preserves the local mass conservation property that is important for subsurface problems. Numerical examples are presented to illustrate the good properties of the proposed approach. If offline spaces are appropriately selected, one can achieve good accuracy with only a few basis functions per coarse element according to the numerical results.A one-dimensional full-range two-phase model to efficiently compute bifurcation diagrams in sub-cooled boiling flows in vertical heated tube.https://www.zbmath.org/1453.654402021-02-27T13:50:00+00:00"Medale, Marc"https://www.zbmath.org/authors/?q=ai:medale.marc"Cochelin, Bruno"https://www.zbmath.org/authors/?q=ai:cochelin.bruno"Bissen, Edouard"https://www.zbmath.org/authors/?q=ai:bissen.edouard"Alpy, Nicolas"https://www.zbmath.org/authors/?q=ai:alpy.nicolasSummary: This paper presents a powerful numerical model to compute bifurcation diagrams in liquid-vapor two-phase fluid flows in vertical heated tube. This full range two-phase model is designed to deal with both single phase (purely liquid or purely vapor) and mixed liquid-vapor configurations that span all flow regimes (laminar and turbulent) in forced, mixed and natural convections. The originality of the proposed methodology is to faithfully integrate the implicit highly nonlinear system of governing equations along branches of steady-state solutions. This is performed by means of a continuation algorithm based on the Asymptotic Numerical Method supplemented with Automatic Differentiation. Then, linear stability analyses are performed at various points of interest, enabling to figure out stability limits within the parameter space in natural circulation configurations. Markedly, Hopf bifurcations that indicate limit-cycle occurrences are identified at low and medium void fractions, respectively, showing the added-value of the approach to track density-wave mechanisms and potential failure of standard application of Ledinegg stability criteria on such cases.The structural topology optimisation based on parameterised level-set method in isogeometric analysis.https://www.zbmath.org/1453.653472021-02-27T13:50:00+00:00"Wu, Zijun"https://www.zbmath.org/authors/?q=ai:wu.zijun"Wang, Shuting"https://www.zbmath.org/authors/?q=ai:wang.shutingSummary: The isogeometric analysis (IGA), which establishes a bridge between CAD and CAE, offers a new convenient framework of optimisation. This article develops an approach to apply the non-uniform rational basis splines (NURBS)-based IGA to the topology optimisation using parameterised level set method. Here, the objective function is evaluated according to the basis function of NURBS and the level set function is constructed through collocation points using the compactly supported radial basis function. To evaluate the equivalent strain energy for each element, the level-set function value of every node is calculated from the corresponding value of collocation points. We have compared the numerical result accuracy as well as the time cost of the proposed method, and it turns out to be very promising.An optimal order a posteriori parameter choice strategy with modified Newton iterative scheme for solving nonlinear ill-posed operator equations.https://www.zbmath.org/1453.651272021-02-27T13:50:00+00:00"Pradeep, D."https://www.zbmath.org/authors/?q=ai:pradeep.dhoorjaty-s"Rajan, M. P."https://www.zbmath.org/authors/?q=ai:rajan.m-pSummary: Study of inverse problems are interesting and mathematically challenging due to the fact that in most of the situation they are unstable with respect to perturbations of the data. In this paper to solve such operator equations, we propose a modified form of Gauss-Newton method combined with an a posteriori parameter choice strategy with the inexact data. Convergence and the convergence rate results are proven. We consider both a-priori and a-posteriori choice rule of parameter that guarantees the scheme converges to the exact solution. The theoretical results are illustrated through numerical examples and compared with the standard scheme to demonstrate that the scheme is stable and achieves good computational output. The salient features of our proposed scheme are: 1) convergence analysis and desired convergence rate require only weaker assumptions compared to many assumptions used in the standard scheme in literature; 2) consideration of an adaptive and numerically stable a posteriori parameter strategy that gives the same order of convergence as that of an a priori method; 3) computation of an optimal order regularisation parameter of the order \(O(\delta^{2/3})\) using a discrepancy principle.Recovering elastic inclusions by shape optimization methods with immersed finite elements.https://www.zbmath.org/1453.740732021-02-27T13:50:00+00:00"Guo, Ruchi"https://www.zbmath.org/authors/?q=ai:guo.ruchi"Lin, Tao"https://www.zbmath.org/authors/?q=ai:lin.tao"Lin, Yanping"https://www.zbmath.org/authors/?q=ai:lin.yanpingSummary: This article presents a finite element method on a fixed mesh for solving a group of inverse geometric problems for recovering the material interface of a linear elasticity system. A partially penalized immersed finite element method is used to discretize both the elasticity interface problems and the objective shape functionals accurately regardless of the shape and location of the interface. Explicit formulas for both the velocity fields and the shape derivatives of IFE shape functions are derived on a fixed mesh and they are employed in the shape sensitivity framework through the discretized adjoint method for accurately and efficiently computing the gradients of objective shape functions with respect to the parameters of the interface curve. The shape optimization for solving an inverse geometric problem is therefore accurately reduced to a constrained optimization that can be implemented efficiently within the IFE framework together with a standard optimization algorithm. We demonstrate features and advantages of the proposed IFE-based shape optimization method by several typical inverse geometric problems for linear elasticity systems.A roadmap for discretely energy-stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity.https://www.zbmath.org/1453.652762021-02-27T13:50:00+00:00"Yang, Zhiguo"https://www.zbmath.org/authors/?q=ai:yang.zhiguo"Dong, Suchuan"https://www.zbmath.org/authors/?q=ai:dong.suchuanSummary: We present a framework for devising discretely energy-stable schemes for general dissipative systems based on a generalized auxiliary variable. The auxiliary variable, a scalar number, can be defined in terms of the energy functional by a general class of functions, not limited to the square root function adopted in previous approaches. The current method has another remarkable property: the computed values for the generalized auxiliary variable are guaranteed to be positive on the discrete level, regardless of the time step sizes or the external forces. This property of guaranteed positivity is not available in previous approaches. A unified procedure for treating the dissipative governing equations and the generalized auxiliary variable on the discrete level has been presented. The discrete energy stability of the proposed numerical scheme and the positivity of the computed auxiliary variable have been proved for general dissipative systems. The current method, termed gPAV (generalized Positive Auxiliary Variable), requires only the solution of linear algebraic equations within a time step. With appropriate choice of the operator in the algorithm, the resultant linear algebraic systems upon discretization involve only constant and time-independent coefficient matrices, which only need to be computed once and can be pre-computed. Several specific dissipative systems are studied in relative detail using the gPAV framework. Ample numerical experiments are presented to demonstrate the performance of the method, and the robustness of the scheme at large time step sizes.Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks.https://www.zbmath.org/1453.650212021-02-27T13:50:00+00:00"Karumuri, Sharmila"https://www.zbmath.org/authors/?q=ai:karumuri.sharmila"Tripathy, Rohit"https://www.zbmath.org/authors/?q=ai:tripathy.rohit-k"Bilionis, Ilias"https://www.zbmath.org/authors/?q=ai:bilionis.ilias"Panchal, Jitesh"https://www.zbmath.org/authors/?q=ai:panchal.jitesh-hSummary: Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions, etc. Because of these functional uncertainties, the stochastic parameter space is often high-dimensional, requiring hundreds, or even thousands, of parameters to describe it. This poses an insurmountable challenge to response surface modeling since the number of forward model evaluations needed to construct an accurate surrogate grows exponentially with the dimension of the uncertain parameter space; a phenomenon referred to as the \textit{curse of dimensionality}. State-of-the-art methods for high-dimensional uncertainty propagation seek to alleviate the curse of dimensionality by performing dimensionality reduction in the uncertain parameter space. However, one still needs to perform forward model evaluations that potentially carry a very high computational burden. We propose a novel methodology for high-dimensional uncertainty propagation of elliptic SPDEs which lifts the requirement for a deterministic forward solver. Our approach is as follows. We parameterize the solution of the elliptic SPDE using a deep residual network (ResNet). In a departure from traditional squared residual (SR) based loss function for training the ResNet, we introduce a physics-informed loss function derived from variational principles. Specifically, our loss function is the expectation of the energy functional of the PDE over the stochastic variables. We demonstrate our solver-free approach through various examples where the elliptic SPDE is subjected to different types of high-dimensional input uncertainties. Also, we solve high-dimensional uncertainty propagation and inverse problems.Collective synchronization of the multi-component Gross-Pitaevskii-Lohe system.https://www.zbmath.org/1453.350272021-02-27T13:50:00+00:00"Bao, Weizhu"https://www.zbmath.org/authors/?q=ai:bao.weizhu"Ha, Seung-Yeal"https://www.zbmath.org/authors/?q=ai:ha.seung-yeal"Kim, Dohyun"https://www.zbmath.org/authors/?q=ai:kim.dohyun"Tang, Qinglin"https://www.zbmath.org/authors/?q=ai:tang.qinglinSummary: In this paper, we propose a multi-component Gross-Pitaevskii-Lohe (GPL for brevity) system in which quantum units interact with each other such that collective behaviors can emerge asymptotically. We introduce several sufficient frameworks leading to complete and practical synchronizations in terms of system parameters and initial data. For the modeling of interaction matrices we classify them into three types (fully identical, weakly identical and heterogeneous) and present emergent behaviors correspond to each interaction matrix. More precisely, for the fully identical case in which all components are same, we expect the emergence of the complete synchronization with exponential convergence rate. On the other hand for the remaining two interaction matrices, we can only show that the practical synchronization occurs under well-prepared initial frameworks. For instance, we assume that a coupling strength is sufficiently large and perturbation of an interaction matrix is sufficiently small. Regarding the practical synchronization estimates, due to the possible blow-up of a solution at infinity, we a priori assume that the \(L^4\)-norm of a solution is bounded on any finite time interval. In our analytical estimates, two-point correlation function approach will play a key role to derive synchronization estimates. We also provide several numerical simulations using time splitting Crank-Nicolson spectral method and compare them with our analytical results.Numerical method for solving time-fractional multi-dimensional diffusion equations.https://www.zbmath.org/1453.653792021-02-27T13:50:00+00:00"Prakash, Amit"https://www.zbmath.org/authors/?q=ai:prakash.amit"Kumar, Manoj"https://www.zbmath.org/authors/?q=ai:yadav.manoj-kumar|kumar.manoj.1|kumar.manoj|kumar.manoj.2Summary: The key object of the current paper is to demonstrate a numerical technique to find the solution of fractional multi-dimensional diffusion equations that describe density dynamics in a material undergoing diffusion with the help of fractional variation iteration method (FVIM). Fractional variation iteration method is not confined to the minor parameter as usual perturbation method. This technique provides us analytical solution in the form of a convergent series with easily computable components. The advantage of this method over other method is that it does not require any linearisation, perturbation and restrictive assumptions.Fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations.https://www.zbmath.org/1453.650622021-02-27T13:50:00+00:00"Bai, Zhong-Zhi"https://www.zbmath.org/authors/?q=ai:bai.zhongzhi"Lu, Kang-Ya"https://www.zbmath.org/authors/?q=ai:lu.kang-yaSummary: The discretizations of two- and three-dimensional spatial fractional diffusion equations with the shifted finite-difference formulas of the Grünwald-Letnikov type can result in discrete linear systems whose coefficient matrices are of the form \(D + T\), where \(D\) is a nonnegative diagonal matrix and \(T\) is a block-Toeplitz with Toeplitz-block matrix or a block-Toeplitz with each block being block-Toeplitz with Toeplitz-block matrix. For these discrete spatial fractional diffusion matrices, we construct diagonal and block-circulant with circulant-block splitting preconditioner for the two-dimensional case, and diagonal and block-circulant with each block being block-circulant with circulant-block splitting preconditioner for the three-dimensional case, to further accelerate the convergence rates of Krylov subspace iteration methods, and we analyze the eigenvalue distributions for the corresponding preconditioned matrices. Theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1, and numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the Krylov subspace iteration methods. Moreover, this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant-plus-diagonal preconditioners in both iteration counts and computing times.A mixed quadrature rule blending Lobatto and Gauss-Legendre three-point rule for approximate evaluation of real definite integrals.https://www.zbmath.org/1453.650572021-02-27T13:50:00+00:00"Tripathy, Arun Kumar"https://www.zbmath.org/authors/?q=ai:tripathy.arun-kumar"Dash, Rajani Ballav"https://www.zbmath.org/authors/?q=ai:dash.rajani-ballav"Baral, Amarendra"https://www.zbmath.org/authors/?q=ai:baral.amarendraSummary: In this paper, a mixed quadrature rule of precision seven has been designed by the linear combination of two rules of precision five. The error analysis of these two formulas has been incorporated. Through some numerical examples the effectiveness of the mixed quadrature rule over its constituent ones has been shown.Decoupled, non-iterative, and unconditionally energy stable large time stepping method for the three-phase Cahn-Hilliard phase-field model.https://www.zbmath.org/1453.762232021-02-27T13:50:00+00:00"Zhang, Jun"https://www.zbmath.org/authors/?q=ai:zhang.jun.5|zhang.jun.2|zhang.jun.1|zhang.jun.8|zhang.jun.3|zhang.jun.9|zhang.jun|zhang.jun.10|zhang.jun.4|zhang.jun.7|zhang.jun.6"Yang, Xiaofeng"https://www.zbmath.org/authors/?q=ai:yang.xiaofengSummary: In this paper, we consider numerical approximations for a three-phase phase-field model, where three fourth-order Cahn-Hilliard equations are nonlinearly coupled together through a Lagrange multiplier term and a sixth-order polynomial bulk potential. By combining the recently developed SAV approach with the linear stabilization technique, we arrive at a novel stabilized-SAV scheme. At each time step, the scheme requires solving only four linear biharmonic equations with constant coefficients, making it the first, to the best of the author's knowledge, totally decoupled, second-order accurate, linear, and unconditionally energy stable scheme for the model. We further prove the unconditional energy stability rigorously and demonstrate the stability and the accuracy of the scheme numerically through the comparisons with the non-stabilized SAV scheme for simulating numerous benchmark numerical examples in 2D and 3D.Compact operators for existence of solution and projection method with multi-wavelet bases to solve (F.IES) and error analysis in Sobolev space.https://www.zbmath.org/1453.654612021-02-27T13:50:00+00:00"Rabbani, Mohsen"https://www.zbmath.org/authors/?q=ai:rabbani.mohsenSummary: In this article, we discuss the existence of solution using compact operators for Fredholm integral equations system (F.IES). Also we use Galerkin method in an \(n\)-dimensional Hilbert space to solve this problem. To reduce the computational operations, we use orthonormal multi-wavelet bases which are constructed by Chebyshev polynomials. Since Chebyshev multi-wavelet bases functions are orthonormal, they help us decrease the operations used in discretizing the integral equations system to an algebraic equations system. The algebraic system contains sparse matrices and it leads to a numerical solution with a high accuracy. Also we consider error analysis in a Sobolev space. Finally for validity and applicability of the above proposed method we compare our results with homotopy perturbation method and Taylor-series expansion method.A moment limiter for the discontinuous Galerkin method on unstructured tetrahedral meshes.https://www.zbmath.org/1453.653182021-02-27T13:50:00+00:00"Giuliani, Andrew"https://www.zbmath.org/authors/?q=ai:giuliani.andrew"Krivodonova, Lilia"https://www.zbmath.org/authors/?q=ai:krivodonova.liliaSummary: We propose a moment limiter for the second order discontinuous Galerkin method on unstructured meshes of tetrahedra. We provide a systematic way for reconstructing and limiting the solution moments from the cell averages. Our analysis, which is based on the linear advection equation, reveals a restriction on the time step and local intervals in which each moment must belong such that a local maximum principle is satisfied. The time step restriction is based on a new measure of cell size that is approximately twice as large as the radius of the inscribed sphere that is typically used. Finally, we discuss limiting across reflecting boundaries for the Euler equations. The efficacy of our limiting algorithm is demonstrated with a number of test problems.On the symmetry properties of a random passive scalar with and without boundaries, and their connection between hot and cold states.https://www.zbmath.org/1453.601162021-02-27T13:50:00+00:00"Camassa, Roberto"https://www.zbmath.org/authors/?q=ai:camassa.roberto"Kilic, Zeliha"https://www.zbmath.org/authors/?q=ai:kilic.zeliha"McLaughlin, Richard M."https://www.zbmath.org/authors/?q=ai:mclaughlin.richard-mSummary: We consider the evolution of a decaying passive scalar in the presence of a Gaussian white noise fluctuating linear shear flow known as the Majda Model. We focus on deterministic initial data and establish the short, intermediate, and long time symmetry properties of the evolving point wise probability measure (PDF) for the random passive scalar. We identify, for the cases of both point source and line source initial data, regions in the x-yplane outside of which the PDF skewness is sign definite for all time, while inside these regions we observe multiple sign changes corresponding to exchanges in symmetry between hot and cold leaning states using exact representation formula for the PDF at the origin, and away from the origin, using numerical evaluation of the exact available Mehler kernel formulae for the scalar's statistical moments. A new, rapidly convergent Monte-Carlo method is developed, dubbed Direct Monte-Carlo (DMC), using the available random Green's functions which allows for the fast construction of the PDF for single point statistics, as well as multi-point statistics including spatially integrated quantities natural for full Monte-Carlo simulations of the underlying stochastic differential equations (FMC). This new method demonstrates the full evolution of the PDF from short times, to its long time, limiting and collapsing universal distribution at arbitrary points in the plane. Further, this method provides a strong benchmark for FMC and we document numbers of field realization criteria for the FMC to faithfully compute this complete dynamics. Armed with this benchmark, we apply the FMC to a channel with a no-flux boundary condition enforced on parallel planes and observe a dramatically different long time state resulting from the existence of the wall. In particular, the channel case collapsing invariant measure has \textit{negative} skewness, with random states leaning heavily towards the hot state, in stark contrast to free space, where the limiting skewness is positive, with its states leaning heavily towards the cold state.Second-order semi-implicit projection methods for micromagnetics simulations.https://www.zbmath.org/1453.820932021-02-27T13:50:00+00:00"Xie, Changjian"https://www.zbmath.org/authors/?q=ai:xie.changjian"García-Cervera, Carlos J."https://www.zbmath.org/authors/?q=ai:garcia-cervera.carlos-j"Wang, Cheng"https://www.zbmath.org/authors/?q=ai:wang.cheng.2|wang.cheng|wang.cheng.1|wang.cheng.3|wang.cheng.4|wang.cheng.5"Zhou, Zhennan"https://www.zbmath.org/authors/?q=ai:zhou.zhennan"Chen, Jingrun"https://www.zbmath.org/authors/?q=ai:chen.jingrun.1Summary: Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on the second-order backward differentiation formula and the second-order interpolation formula using the information at previous two temporal steps. Unconditional unique solvability of both methods is proved, with their second-order accuracy verified through numerical examples in both 1D and 3D. The efficiency of both methods is compared to that of another two popular methods. In addition, we test the robustness of both methods for the first benchmark problem with a ferromagnetic thin film material from National Institute of Standards and Technology.A nonlinear moment model for radiative transfer equation in slab geometry.https://www.zbmath.org/1453.652482021-02-27T13:50:00+00:00"Fan, Yuwei"https://www.zbmath.org/authors/?q=ai:fan.yuwei"Li, Ruo"https://www.zbmath.org/authors/?q=ai:li.ruo"Zheng, Lingchao"https://www.zbmath.org/authors/?q=ai:zheng.lingchaoSummary: This paper is concerned with the approximation of the radiative transfer equation for a grey medium in the slab geometry by the moment method. We develop a novel moment model inspired by the classical \(P_N\) model and \(M_N\) model. The new model takes the ansatz of the \(M_1\) model as the weight function and follows the primary idea of the \(P_N\) model to approximate the specific intensity by expanding it around the weight function in terms of orthogonal polynomials. The weight function uses the information of the first two moments, which brings the new model the capability to approximate an anisotropic distribution. Mathematical properties of the moment model are investigated, and particularly the hyperbolicity and the characteristic structure of the Riemann problem of the model with three moments are studied in detail. Some numerical simulations demonstrate its numerical efficiency and show its superior in comparison to the \(P_N\) model.On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations.https://www.zbmath.org/1453.850022021-02-27T13:50:00+00:00"Dumbser, Michael"https://www.zbmath.org/authors/?q=ai:dumbser.michael"Fambri, Francesco"https://www.zbmath.org/authors/?q=ai:fambri.francesco"Gaburro, Elena"https://www.zbmath.org/authors/?q=ai:gaburro.elena"Reinarz, Anne"https://www.zbmath.org/authors/?q=ai:reinarz.anneSummary: In this paper we propose an extension of the generalized Lagrangian multiplier method (GLM) of [\textit{C. D. Munz} et al., ibid. 161, No. 2, 484--511 (2000; Zbl 0970.78010); \textit{A. Dedner} et al., ibid. 175, No. 2, 645--673 (2002; Zbl 1059.76040)], which was originally conceived for the numerical solution of the Maxwell and MHD equations with divergence-type involutions, to the case of hyperbolic PDE systems with curl-type involutions. The key idea here is to solve an \textit{augmented} PDE system, in which curl errors propagate away via a Maxwell-type evolution system. The new approach is first presented on a simple model problem, in order to explain the basic ideas. Subsequently, we apply it to a strongly hyperbolic first order reduction of the CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, which is endowed with 11 curl constraints. Several numerical examples, including the long-time evolution of a stable neutron star in anti-Cowling approximation, are presented in order to show the obtained improvements with respect to the standard formulation without special treatment of the curl involution constraints. The main advantages of the proposed GLM approach are its complete independence of the underlying numerical scheme and grid topology and its easy implementation into existing computer codes. However, this flexibility comes at the price of needing to add for each curl involution one additional 3 vector plus another scalar in the augmented system for homogeneous curl constraints, and even two additional scalars for non-homogeneous curl involutions. For the FO-CCZ4 system with 11 homogeneous curl involutions, this means that additional 44 evolution quantities need to be added.An efficient class of WENO schemes with adaptive order for unstructured meshes.https://www.zbmath.org/1453.652082021-02-27T13:50:00+00:00"Balsara, Dinshaw S."https://www.zbmath.org/authors/?q=ai:balsara.dinshaw-s"Garain, Sudip"https://www.zbmath.org/authors/?q=ai:garain.sudip-k"Florinski, Vladimir"https://www.zbmath.org/authors/?q=ai:florinski.vladimir"Boscheri, Walter"https://www.zbmath.org/authors/?q=ai:boscheri.walterSummary: Recent advances in finite-difference WENO schemes for hyperbolic conservation laws have resulted in WENO schemes with adaptive order of accuracy. For instance, a WENO-AO(5,3) scheme can provide up to fifth order of accuracy when the smoothness of the solution in the fifth order stencil warrants it, and yet, it can adaptively drop down to third order of accuracy when the higher order is not warranted by the solution on the mesh. Having an analogous capability for finite-volume WENO schemes for hyperbolic conservation laws, especially on unstructured meshes, can be very valuable. The present paper documents the design of finite volume WENO-AO(4,3) and WENO-AO(5,3) schemes for unstructured meshes. As with WENO-AO for structured meshes, the key advance lies in realizing that there is a favorable basis set, which is very easily constructed, and in which the computation is dramatically simplified. As with finite-difference WENO, we realize that one can make a non-linear hybridization between a large, centered, very high accuracy stencil and a lower order central WENO scheme that is, nevertheless, very stable and capable of capturing physically meaningful extrema. This yields a class of adaptive order WENO schemes that work well on unstructured meshes. On both the large and small stencils we have been able to make the stencil evaluation step very efficient owing to the choice of a favorable Taylor series basis. By extending the Parallel Axis Theorem, we show that there is a significant simplification in the finite volume reconstruction. Instead of solving a constrained least squares problem, our method only requires the solution of a smaller least squares problem on each stencil. This also simplifies the matrix assembly and solution for each stencil. The evaluation of smoothness indicators is also simplified. Accuracy tests show that the method meets its design accuracy. Several stringent test problems are presented to demonstrate that the method works very robustly and very well. The test problems are chosen to show that our method can be applied to many different meshes that are used to map geometric complexity or solution complexity.What is the fractional Laplacian? A comparative review with new results.https://www.zbmath.org/1453.351792021-02-27T13:50:00+00:00"Lischke, Anna"https://www.zbmath.org/authors/?q=ai:lischke.anna"Pang, Guofei"https://www.zbmath.org/authors/?q=ai:pang.guofei"Gulian, Mamikon"https://www.zbmath.org/authors/?q=ai:gulian.mamikon"Song, Fangying"https://www.zbmath.org/authors/?q=ai:song.fangying"Glusa, Christian"https://www.zbmath.org/authors/?q=ai:glusa.christian"Zheng, Xiaoning"https://www.zbmath.org/authors/?q=ai:zheng.xiaoning"Mao, Zhiping"https://www.zbmath.org/authors/?q=ai:mao.zhiping"Cai, Wei"https://www.zbmath.org/authors/?q=ai:cai.wei"Meerschaert, Mark M."https://www.zbmath.org/authors/?q=ai:meerschaert.mark-m"Ainsworth, Mark"https://www.zbmath.org/authors/?q=ai:ainsworth.mark"Karniadakis, George Em"https://www.zbmath.org/authors/?q=ai:karniadakis.george-emSummary: The fractional Laplacian in \(\mathbb{R}^d\), which we write as \((- \Delta)^{\alpha / 2}\) with \(\alpha \in(0, 2)\), has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: ``What is the fractional Laplacian?'' Beginning from first principles, we compare several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties. Then, we present quantitative comparisons using a sample of state-of-the-art methods. We discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems: radial basis function collocation (for the Riesz fractional Laplacian) and nonharmonic lifting (for the spectral fractional Laplacian). In our numerical studies, we aim to compare different definitions on bounded domains using a collection of benchmark problems. We consider the fractional Poisson equation with both zero and nonzero boundary conditions, where the fractional Laplacian is defined according to the Riesz definition, the spectral definition, the directional definition, and the horizon-based nonlocal definition. We verify the accuracy of the numerical methods used in the approximations for each operator, and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.An eight-order accurate numerical method for the solution of 2D Helmholtz equation.https://www.zbmath.org/1453.653862021-02-27T13:50:00+00:00"Tadi, M."https://www.zbmath.org/authors/?q=ai:tadi.mohsenSummary: This note is concerned with a numerical method for the solution of 2D Helmholtz equation in unit square. The method uses a finite difference approximation in one coordinate space. Similar to the method of line, the method treats the working equation as a system of ordinary differential equation in the remaining independent variable. The method uses a coordinate transformation to decouple the system of ODE. Using this procedure, it is possible to formulate numerical schemes with arbitrary orders of accuracy. Numerical results for an eight-order accurate are presented.Corrigendum to: ``Application of the natural stress formulation for solving unsteady viscoelastic contraction flows''.https://www.zbmath.org/1453.761272021-02-27T13:50:00+00:00"Evans, Jonathan D."https://www.zbmath.org/authors/?q=ai:evans.jonathan-david|evans.jonathan-d"França, Hugo L."https://www.zbmath.org/authors/?q=ai:franca.hugo-l"Oishi, Cassio M."https://www.zbmath.org/authors/?q=ai:oishi.cassio-mThis is a correction to the authors' paper [ibid. 388, 462--489 (2019; Zbl 1452.76152)].Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs.https://www.zbmath.org/1453.652802021-02-27T13:50:00+00:00"Dektor, Alec"https://www.zbmath.org/authors/?q=ai:dektor.alec"Venturi, Daniele"https://www.zbmath.org/authors/?q=ai:venturi.danieleSummary: We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets. This process, which can be conveniently visualized in terms of binary trees, yields series expansions analogous to the classical Tensor-Train and Hierarchical Tucker tensor formats. By enforcing dynamic orthogonality conditions at each level of the binary tree, we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition. This allows us to effectively compute the solution to high-dimensional time-dependent nonlinear PDEs on tensor manifolds of constant rank, with no need for rank reduction methods. We also propose new algorithms for dynamic addition and removal of modes within each subspace. Numerical examples are presented and discussed for high-dimensional hyperbolic and parabolic PDEs in bounded domains.A comparative study of finite element method for the numerical solution of reaction-diffusion problem.https://www.zbmath.org/1453.653362021-02-27T13:50:00+00:00"Maharana, Nini"https://www.zbmath.org/authors/?q=ai:maharana.nini"Nayak, Ameeya Kumar"https://www.zbmath.org/authors/?q=ai:nayak.ameeya-kumarSummary: This paper presents a least-squares finite element approach for the solution of reaction-diffusion pellet problem for slab geometry by considering some special types of reactions which are applicable for chemical transformation of natural gas components. The nonlinear terms are treated with Newton's linearisation technique. A detailed comparison with other weighted residual methods such as collocation, sub-domain and Galerkin methods are also examined. The issues related to chemical reactor problems and their error analysis is discussed both theoretically and numerically. The least-squares method suffers convergency effect by using the Picard's method instead of Newton's iteration method as obtained by Galerkin and orthogonal collocation methods.A robust CFL condition for the discontinuous Galerkin method on triangular meshes.https://www.zbmath.org/1453.652672021-02-27T13:50:00+00:00"Chalmers, N."https://www.zbmath.org/authors/?q=ai:chalmers.noel"Krivodonova, L."https://www.zbmath.org/authors/?q=ai:krivodonova.liliaSummary: When the discontinuous Galerkin (DG) method is applied to hyperbolic problems in two dimensions on triangular meshes and paired with an explicit time integration scheme, an exact CFL condition is not known. The stability condition which is most usually implemented involves scaling the time step by the smallest radius of the inscribed circle in every cell. However, this is known to not provide a tight bound on the largest possible stable time step in some cases. In this paper, we apply the DG method to a simple linear problem and derive a PDE which is satisfied by the numerical solution itself. By applying classical Fourier analysis to the solutions of this PDE we find a natural scaling of the spectrum of the DG spatial operator by a parameter \(h_j\), which can be seen to be the width of the cell \(\Omega_j\) along the characteristic direction of flow. We use this parameter to propose a new CFL condition and show through several numerical examples that we are able to select significantly larger time steps than usually obtained using the inscribed radii of the computational cells.A novel multi-subpopulation cooperative particle swarm optimisation.https://www.zbmath.org/1453.902022021-02-27T13:50:00+00:00"Lin, Guohan"https://www.zbmath.org/authors/?q=ai:lin.guohan"Zhang, Jing"https://www.zbmath.org/authors/?q=ai:zhang.jing.8|zhang.jing|zhang.jing.9|zhang.jing.12|zhang.jing.7|zhang.jing.5|zhang.jing.1|zhang.jing.3|zhang.jing.2|zhang.jing.11|zhang.jing.6|zhang.jing.10Summary: The basic particle swarm optimisation (PSO) algorithm is easily trapped in local optima. To deal with this problem, a multi-subpopulation cooperative particle swarm optimisation (MCPSO) is presented. In the proposed algorithm, the particles are divided into several normal subpopulations and an elite subpopulation. The selected individuals in normal subpopulation are memorised into the elite subpopulation, and some individuals in normal subpopulation are replaced by the best particles from the elite subpopulation. Different subpopulation adopts different evolution model. This strategy can maintain the diversity of the population and avoid the premature convergence. The performance of the proposed algorithm is evaluated by testing on standard benchmark functions. The experimental results show that the proposed algorithm has better convergent rate and high solution accuracy.Sequential local mesh refinement solver with separate temporal and spatial adaptivity for non-linear two-phase flow problems.https://www.zbmath.org/1453.760762021-02-27T13:50:00+00:00"Li, Hanyu"https://www.zbmath.org/authors/?q=ai:li.hanyu"Leung, Wing Tat"https://www.zbmath.org/authors/?q=ai:leung.wingtat"Wheeler, Mary F."https://www.zbmath.org/authors/?q=ai:wheeler.mary-fanettSummary: Convergence failure and slow convergence rates are among the biggest challenges with solving the system of non-linear equations numerically. Although mitigated, such issues still linger when using strictly small time steps and unconditionally stable fully implicit schemes. The price that comes with restricting time steps to small scales is the enormous computational load, especially in large-scale models. To address this problem, we introduce a sequential local mesh refinement framework of temporal and spatial adaptivity to optimize convergence rate and prevent convergence failure, while not restricting the whole system to small time steps, thus improving computational efficiency. Two types of error estimators are introduced to estimate the spatial discretization error, the temporal discretization error separately. These estimators provide a global upper bounds on the dual norm of the residual and the non-conformity of the numerical solution for non-linear two phase flow models. The mesh refinement algorithm starts from solving the problem on the coarsest space-time mesh, then the mesh is refined sequentially based on the spatial error estimator and the temporal error estimator. After each refinement, the solution from the previous mesh is used to estimate the initial guess of unknowns on the current mesh for faster convergence. Numerical results are presented to confirm accuracy of our algorithm as compared to the uniformly fine time step and fine spatial discretization solution. We observe around 25 times speedup in the solution time by using our algorithm.Eigenvalues bounds for symmetric interval matrices.https://www.zbmath.org/1453.150112021-02-27T13:50:00+00:00"Singh, Sukhjit"https://www.zbmath.org/authors/?q=ai:singh.sukhjit"Gupta, D. K."https://www.zbmath.org/authors/?q=ai:gupta.dharmendra-kumarSummary: The aim of this paper is to approximate the eigenvalues bounds of real symmetric interval matrices as sharp as possible. Using the concepts of interval analysis, this is done by solving the standard interval eigenvalue problem by applying the interval extension of the single step eigen perturbation method. The deviation amplitude of the interval matrix is considered as a perturbation around the nominal value of the interval matrix. The mathematical formulation of the method is described. Two numerical examples are worked out and the results obtained are compared with the results of existing methods. It is observed that our method is reliable, efficient and gives better results for all examples considered.An approximate numerical solution to the Graetz problem with constant wall temperature.https://www.zbmath.org/1453.800052021-02-27T13:50:00+00:00"Belhocine, Ali"https://www.zbmath.org/authors/?q=ai:belhocine.aliSummary: The present set of themes related to the investigations of heat transfer by convection and the transport phenomenon in a cylindrical pipe in laminar flow, is commonly called the Graetz problem, which is to explore the evolution of the temperature profile for a fluid flow in fully developed laminar flow. A numerical method was developed in this work, for visualisation of the temperature profile in the fluid flow, whose strategy of calculation is based on the orthogonal collocation method followed by the finite difference method (Crank-Nicholson method). The calculations were effected through a FORTRAN computer program and the results show that orthogonal collocation method giving better results than Crank-Nicholson method.Global optimization for data assimilation in landslide tsunami models.https://www.zbmath.org/1453.860282021-02-27T13:50:00+00:00"Ferreiro-Ferreiro, A. M."https://www.zbmath.org/authors/?q=ai:ferreiro-ferreiro.ana-m"García-Rodríguez, J. A."https://www.zbmath.org/authors/?q=ai:garcia-rodriguez.jose-antonio"López-Salas, J. G."https://www.zbmath.org/authors/?q=ai:lopez-salas.jose-german"Escalante, C."https://www.zbmath.org/authors/?q=ai:escalante.c"Castro, M. J."https://www.zbmath.org/authors/?q=ai:castro.manuel-jSummary: The goal of this article is to make automatic data assimilation for a landslide tsunami model, given by the coupling between a non-hydrostatic multi-layer shallow-water and a Savage-Hutter granular landslide model for submarine avalanches. The coupled model is discretized using a positivity preserving second-order path-conservative finite volume scheme. Then, the data assimilation problem is posed in a global optimization framework. Later, multi-path parallel metaheuristic stochastic global optimization algorithms are developed. More precisely, a multi-path Simulated Annealing algorithm is compared with a multi-path hybrid global optimization algorithm based on coupling Simulated Annealing with gradient local searchers.Time adaptive conservative finite volume method.https://www.zbmath.org/1453.652532021-02-27T13:50:00+00:00"Jenny, Patrick"https://www.zbmath.org/authors/?q=ai:jenny.patrickSummary: Finite volume methods for time dependent problems typically employ time integration schemes with constant time step sizes. The latter, in order to ensure stability and time accurate solutions, have to be chosen small enough, such that the Courant-Friedrichs-Lewy (CFL) number stays below a critical value everywhere. In many cases, however, this time step size limitation leads to huge numbers of very small time steps, which renders simulations very expensive. Motivated by this drawback of conventional time stepping schemes, various sub-time stepping algorithms have been proposed. However, most of them are inherently asynchronous, require small local CFL numbers or are not strictly conservative. In this paper a new adaptive time integration scheme for finite volume methods, which is conservative, of high spatial and temporal order, robust and easy to implement is presented. It relies on local sub-time steps which are fractions of a global time step by powers of two, i.e., the grid cells proceed asynchronously in the order of their termination time, but since they all synchronize at the end of each global time step, it is possible to guarantee continuity of the mean fluxes and thus strict conservation at the global time step resolution. Numerical experiments with 1D and 2D test cases demonstrate that the adaptive conservative time integration (ACTI) scheme can achieve extreme speed-up factors over conventional time integration, while still maintaining high spatial and temporal accuracy.Symplectic integration with non-canonical quadrature for guiding-center orbits in magnetic confinement devices.https://www.zbmath.org/1453.654362021-02-27T13:50:00+00:00"Albert, Christopher G."https://www.zbmath.org/authors/?q=ai:albert.christopher-g"Kasilov, Sergei V."https://www.zbmath.org/authors/?q=ai:kasilov.sergei-v"Kernbichler, Winfried"https://www.zbmath.org/authors/?q=ai:kernbichler.winfriedSummary: We study symplectic numerical integration of mechanical systems with a Hamiltonian specified in non-canonical coordinates and its application to guiding-center motion of charged plasma particles in magnetic confinement devices. The technique combines time-stepping in canonical coordinates with quadrature in non-canonical coordinates and is applicable in systems where a global transformation to canonical coordinates is known explicitly but its inverse is not. A fully implicit class of symplectic Runge-Kutta schemes has recently been introduced and applied to guiding-center motion by \textit{R. Zhang} et al. [``Canonicalization and symplectic simulation of the gyrocenter dynamics in time-independent magnetic fields'', Phys. Plasmas 21, No. 3, Article ID 032504, 12 p. (2014; \url{doi:10.1063/1.4867669})]. Here a generalization of this approach with emphasis on semi-implicit partitioned schemes is described together with methods to enhance performance, in particular avoiding evaluation of non-canonical variables at full time steps. For application in toroidal plasma confinement configurations with nested magnetic flux surfaces a global canonicalization of coordinates for the guiding-center Lagrangian by a spatial transform is presented that allows for pre-computation of the required map in a parallel algorithm in the case of time-independent magnetic field geometry. Guiding-center orbits are studied in stationary magnetic equilibrium fields of an axisymmetric tokamak and a realistic three-dimensional stellarator configuration. Superior long-term properties of symplectic methods are demonstrated in comparison to a conventional adaptive Runge-Kutta scheme. Finally statistics of fast fusion alpha particle losses over their slowing-down time are computed in the stellarator field on a representative sample, reaching a speed-up of the symplectic Euler scheme by more than a factor three compared to usual Runge-Kutta schemes while keeping the same statistical accuracy and linear scaling with the number of computing threads.A hybrid high-order method for a coupled Stokes-Darcy problem on general meshes.https://www.zbmath.org/1453.760852021-02-27T13:50:00+00:00"Zhang, Yongchao"https://www.zbmath.org/authors/?q=ai:zhang.yongchao"Mei, Liquan"https://www.zbmath.org/authors/?q=ai:mei.liquan"Li, Rui"https://www.zbmath.org/authors/?q=ai:li.rui.2|li.rui.4|li.rui.3|li.rui|li.rui.1Summary: In this work, a hybrid high-order (HHO) method on general meshes is presented to solve a coupled Stokes-Darcy problem with the Beavers-Joseph-Saffman interface condition. Constructed on polynomials of arbitrary degree \(k \geq 0\), the numerical method is established in terms of discrete unknowns attached to mesh faces and cells (or elements). The unified discrete scheme for Stokes equation and Darcy equation is given by the continuity condition of the interface. The unique solvability of the discrete scheme is proved. Moreover, the energy error estimate for the velocity and \(L^2\)-error estimate for pressure of order \((k + 1)\) are derived. Finally, a series of numerical experiments are reported to illustrate the accuracy, mass conservation and robustness of our method.A low-rank projector-splitting integrator for the Vlasov-Maxwell equations with divergence correction.https://www.zbmath.org/1453.653572021-02-27T13:50:00+00:00"Einkemmer, Lukas"https://www.zbmath.org/authors/?q=ai:einkemmer.lukas"Ostermann, Alexander"https://www.zbmath.org/authors/?q=ai:ostermann.alexander"Piazzola, Chiara"https://www.zbmath.org/authors/?q=ai:piazzola.chiaraSummary: The Vlasov-Maxwell equations are used for the kinetic description of magnetized plasmas. As they are posed in an up to \(3 + 3\) dimensional phase space, solving this problem is extremely expensive from a computational point of view. In this paper, we exploit the low-rank structure in the solution of the Vlasov equation. More specifically, we consider the Vlasov-Maxwell system and propose a dynamic low-rank integrator. The key idea is to approximate the dynamics of the system by constraining it to a low-rank manifold. This is accomplished by a projection onto the tangent space. There, the dynamics is represented by the low-rank factors, which are determined by solving lower-dimensional partial differential equations. The proposed scheme performs well in numerical experiments and succeeds in capturing the main features of the plasma dynamics. We demonstrate this good behavior for a range of test problems. The coupling of the Vlasov equation with the Maxwell system, however, introduces additional challenges. In particular, the divergence of the electric field resulting from Maxwell's equations is not consistent with the charge density computed from the Vlasov equation. We propose a correction based on Lagrange multipliers which enforces Gauss' law up to machine precision.Numerical computation of integral of analytic functions in complex plane.https://www.zbmath.org/1453.650512021-02-27T13:50:00+00:00"Hota, Manoj Kumar"https://www.zbmath.org/authors/?q=ai:hota.manoj-kumar"Dash, Biranchi Narayan"https://www.zbmath.org/authors/?q=ai:dash.biranchi-narayan"Mohanty, Prasanta Kumar"https://www.zbmath.org/authors/?q=ai:mohanty.prasanta-kumarSummary: A family of interpolatory type of quadrature rules of degree of precision at least seven has been constructed for the approximate evaluation of contour integrals of analytic functions along a directed line segment in the complex plane. The relative accuracies of rules have been studied in their respective classes and also numerically verified by integrating some standard test integrals.A discontinuous Galerkin method for the simulation of compressible gas-gas and gas-water two-medium flows.https://www.zbmath.org/1453.760632021-02-27T13:50:00+00:00"Cheng, Jian"https://www.zbmath.org/authors/?q=ai:cheng.jian"Zhang, Fan"https://www.zbmath.org/authors/?q=ai:zhang.fan"Liu, Tiegang"https://www.zbmath.org/authors/?q=ai:liu.tiegangSummary: In this paper, we develop a new discontinuous Galerkin method for the simulation of shocks and interfaces in compressible gas-gas and gas-water two-medium flows by solving the \(\gamma \)-based model. The spatial discretization is carefully designed to possess the following features: discrete conservation in terms of the total mass, total momentum and total energy; high-order accuracy and consistency for smooth flows; free of oscillations at an isolated material interface. In order to handle potential discontinuities arising in the simulation, a nonlinear limiter based on the weighted essentially non-oscillatory (WENO) strategy is employed to suppress numerical oscillations and to preserve high order of accuracy in regions of smooth flows. The WENO reconstruction is imposed on suitably selected quantities, rather than the conserved ones. In case for discontinuities with large pressure ratio, low density and dramatic change of material property where unphysical variables may be encountered, a posteriori solution correction on the subcell level is locally adopted to enhance the robustness. A series of typical test cases for both one- and two-dimensional problems are provided to demonstrate the performance of the proposed method.Numerical methods for Bogoliubov-de Gennes excitations of Bose-Einstein condensates.https://www.zbmath.org/1453.810652021-02-27T13:50:00+00:00"Gao, Yali"https://www.zbmath.org/authors/?q=ai:gao.yali"Cai, Yongyong"https://www.zbmath.org/authors/?q=ai:cai.yongyongSummary: In this paper, we study the analytical properties and the numerical methods for the Bogoliubov-de Gennes equations (BdGEs) describing the elementary excitation of Bose-Einstein condensates around the mean field ground state, which is governed by the Gross-Pitaevskii equation (GPE). Derived analytical properties of BdGEs can serve as benchmark tests for numerical algorithms and three numerical methods are proposed to solve the BdGEs, including sine-spectral method, central finite difference method and compact finite difference method. Extensive numerical tests are provided to validate the algorithms and confirm that the sine-spectral method has spectral accuracy in spatial discretization, while the central finite difference method and the compact finite difference method are second-order and fourth-order accurate, respectively. Finally, sine-spectral method is extended to study elementary excitations under the optical lattice potential and solve the BdGEs around the first excited states of the GPE. The numerical experiments demonstrate the efficiency and accuracy of the proposed methods for solving BdGEs.Assessing an efficient hybrid of Monte Carlo technique (GSA-GLUE) in uncertainty and sensitivity analysis of vanGenuchten soil moisture characteristics curve.https://www.zbmath.org/1453.860392021-02-27T13:50:00+00:00"Etminan, Samaneh"https://www.zbmath.org/authors/?q=ai:etminan.samaneh"Jalali, Vahidreza"https://www.zbmath.org/authors/?q=ai:jalali.vahidreza"Mahmoodabadi, Majid"https://www.zbmath.org/authors/?q=ai:mahmoodabadi.majid"siuki, Abbas Khashei"https://www.zbmath.org/authors/?q=ai:siuki.abbas-khashei"Bilondi, Mohsen Pourreza"https://www.zbmath.org/authors/?q=ai:bilondi.mohsen-pourrezaSummary: Studying model uncertainty and identifying the parameter uncertainty in the modeling of water flow through the soil is useful to improve water and soil management. This research aimed to assess the uncertainty of the parameters of soil water retention curve (SWRC) models using an efficient hybrid of the Monte Carlo technique e.g. generalized likelihood uncertainty estimation (GLUE). GLUE estimates the parameters of vanGenuchten, vanGenuchten-Mualem, and vanGenuchten-Burdine models for four soil classes. Also, to evaluate the relative importance of the model parameters, generalized sensitivity analysis (GSA) was performed. The results of the uncertainty analysis showed that among the studied models, the vanGenuchten-Mualem model with the indices of S = 0.05, T = 0.4, \textit{d-factor} = 0.25 and, \(P_{\mathrm{CI}} = 100\) was considered as the most accurate model with the least uncertainty. Also, the results of GSA were demonstrated that alpha and n parameters were sensitive parameters in the models. Consequently, identifying the uncertainty of the SWRC model structure and its parameters, relevant models with higher accuracy can be used in the study of soil water processes, and better water resource allocation.An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: formulation, analysis, algorithm, and simulation.https://www.zbmath.org/1453.654052021-02-27T13:50:00+00:00"Domínguez, V."https://www.zbmath.org/authors/?q=ai:dominguez.victor"Ganesh, M."https://www.zbmath.org/authors/?q=ai:ganesh.madhu|ganesh.mahadevan"Sayas, F. J."https://www.zbmath.org/authors/?q=ai:sayas.francisco-javierSummary: A natural medium for wave propagation comprises a coupled bounded heterogeneous region and an unbounded homogeneous free-space. Frequency-domain wave propagation models in the medium, such as the variable coefficient Helmholtz equation, include a faraway decay radiation condition (RC). It is desirable to develop algorithms that incorporate the full physics of the heterogeneous and unbounded medium wave propagation model, and avoid an approximation of the RC. In this work we first present and analyze an overlapping decomposition framework that is equivalent to the full-space heterogeneous-homogenous continuous model, governed by the Helmholtz equation with a spatially dependent refractive index and the RC. Our novel overlapping framework allows the user to choose two free boundaries, and gain the advantage of applying established high-order finite and boundary element methods (FEM and BEM) to simulate an equivalent coupled model. The coupled model comprises auxiliary interior bounded heterogeneous and exterior unbounded homogeneous Helmholtz problems. A smooth boundary can be chosen for simulating the exterior problem using a spectrally accurate BEM, and a simple boundary can be used to develop a high-order FEM for the interior problem. Thanks to the spectral accuracy of the exterior computational model, the resulting coupled system in the overlapping region is relatively very small. Using the decomposed equivalent framework, we develop a novel overlapping FEM-BEM algorithm for simulating the acoustic or electromagnetic wave propagation in two dimensions. Our FEM-BEM algorithm for the full-space model incorporates the RC exactly. Numerical experiments demonstrate the efficiency of the FEM-BEM approach for simulating smooth and non-smooth wave fields, with the latter induced by a complex heterogeneous medium and a discontinuous refractive index.Optimization of fast algorithms for global quadrature by expansion using target-specific expansions.https://www.zbmath.org/1453.654642021-02-27T13:50:00+00:00"Wala, Matt"https://www.zbmath.org/authors/?q=ai:wala.matt"Klöckner, Andreas"https://www.zbmath.org/authors/?q=ai:klockner.andreasSummary: We develop an algorithm for the asymptotically fast evaluation of layer potentials close to and on the source geometry, combining Geometric Global Accelerated QBX (`GIGAQBX') and target-specific expansions. GIGAQBX is a fast high-order scheme for evaluation of layer potentials based on Quadrature by Expansion (`QBX') using local expansions formed via the Fast Multipole Method (FMM). Target-specific expansions serve to lower the cost of the formation and evaluation of QBX local expansions, reducing the associated computational effort from \(O((p + 1)^2)\) to \(O(p + 1)\) in three dimensions, without any accuracy loss compared with conventional expansions, but with the loss of source/target separation in the expansion coefficients. GIGAQBX is a `global' QBX scheme, meaning that the potential is mediated entirely through expansions for points close to or on the boundary. In our scheme, this single global expansion is decomposed into two parts that are evaluated separately: one part incorporating near-field contributions using target-specific expansions, and one part using conventional spherical harmonic expansions of far-field contributions, noting that convergence guarantees only exist for the sum of the two sub-expansions. By contrast, target-specific expansions were originally introduced as an acceleration mechanism for `local' QBX schemes, in which the far-field does not contribute to the QBX expansion. Compared with the unmodified GIGAQBX algorithm, we show through a reproducible, time-calibrated cost model that the combined scheme yields a considerable cost reduction for the near-field evaluation part of the computation. We support the effectiveness of our scheme through numerical results demonstrating performance improvements for Laplace and Helmholtz kernels.A parallel multithreaded sparse triangular linear system solver.https://www.zbmath.org/1453.650592021-02-27T13:50:00+00:00"Çuğu, İlke"https://www.zbmath.org/authors/?q=ai:cugu.ilke"Manguoğlu, Murat"https://www.zbmath.org/authors/?q=ai:manguoglu.muratSummary: We propose a parallel sparse triangular linear system solver based on the Spike algorithm. Sparse triangular systems are required to be solved in many applications. Often, they are a bottleneck due to their inherently sequential nature. Furthermore, typically many successive systems with the same coefficient matrix and with different right hand side vectors are required to be solved. The proposed solver decouples the problem at the cost of extra arithmetic operations as in the banded case. Compared to the banded case, there are extra savings due to the sparsity of the triangular coefficient matrix. We show the parallel performance of the proposed solver against the state-of-the-art parallel sparse triangular solver in Intel's Math Kernel Library (MKL) on a multicore architecture. We also show the effect of various sparse matrix reordering schemes. Numerical results show that the proposed solver outperforms MKL's solver in \(\sim 80\%\) of cases by a factor of 2.47, on average.An improved optimal eighth-order iterative scheme with its dynamical behaviour.https://www.zbmath.org/1453.650992021-02-27T13:50:00+00:00"Choubey, Neha"https://www.zbmath.org/authors/?q=ai:choubey.neha"Jaiswal, J. P."https://www.zbmath.org/authors/?q=ai:jaiswal.jai-prakashSummary: In this paper, we have designed an improvement of the very recently constructed sixth-order iterative method without memory by using the weight function technique. Our scheme is optimal because it reaches the highest possible order using four function evaluations. Ten nonlinear test functions have been taken into account to check the reliability of the proposed scheme. Finally, dynamics of the proposed methods with some existing methods, illustrated in the paper.Predicting plume spreading during \(\mathrm{CO_2}\) geo-sequestration: benchmarking a new hybrid finite element-finite volume compositional simulator with asynchronous time marching.https://www.zbmath.org/1453.860072021-02-27T13:50:00+00:00"Shao, Qi"https://www.zbmath.org/authors/?q=ai:shao.qi"Matthai, Stephan"https://www.zbmath.org/authors/?q=ai:matthai.stephan-k"Driesner, Thomas"https://www.zbmath.org/authors/?q=ai:driesner.thomas"Gross, Lutz"https://www.zbmath.org/authors/?q=ai:gross.lutzSummary: In this paper, we present the results of benchmark simulations for plume spreading during \(\mathrm{CO_2}\) geo-sequestration conducted with the newly developed Australian \(\mathrm{CO_2}\) Geo-Sequestration Simulator (ACGSS). The simulator uses a hybrid finite element-finite volume (FEFVM) simulation framework, integrating an asynchronous local time stepping method for multi-phase multi-component transport and a novel non-iterative flash calculation approach for the phase equilibrium. The benchmark investigates four standard \(\mathrm{CO_2}\) storage test cases that are widely used to assess the performance of simulation tools for carbon geo-sequestration: (A) radial flow from a \(\mathrm{CO_2}\) injection well; (B) \(\mathrm{CO_2}\) discharge along a fault zone; (C) \(\mathrm{CO_2}\) injection into a layered brine formation; and (D) leakage through an abandoned well. For these applications, ACGSS gives results similar to well-established compositional simulators. Minor discrepancies can be rationalised in terms of the alternative, spatially adaptive discretisation and the treatment of NaCl solubility. While these benchmarks cover issues related to compositional simulation, they do not address the accurate representation of geologically challenging features of \(\mathrm{CO_2}\) storage sites. An additional 3D application scenario of a complexly faulted storage site demonstrates the advantages of the FEFVM discretisation used in the ACGSS for resolving the geometric complexity of geologic storage sites. This example also highlights the significant computational benefits gained from the use of the asynchronous time marching scheme.Compensated \(\theta\)-Milstein methods for stochastic differential equations with Poisson jumps.https://www.zbmath.org/1453.650222021-02-27T13:50:00+00:00"Ren, Quanwei"https://www.zbmath.org/authors/?q=ai:ren.quanwei"Tian, Hongjiong"https://www.zbmath.org/authors/?q=ai:tian.hongjiongSummary: In this paper, we are concerned with numerical methods for solving stochastic differential equations with Poisson-driven jumps. We construct a class of compensated \(\theta\)-Milstein methods and study their mean-square convergence and asymptotic mean-square stability. Sufficient and necessary conditions for the asymptotic mean-square stability of the compensated \(\theta\)-Milstein methods when applied to a scalar linear test equation are derived. We compare the asymptotic mean-square stability region of the linear test equation with that of the compensated \(\theta \)-Milstein methods with different \(\theta\) values. Numerical results are given to verify our theoretical results.Long-term analysis of stochastic \(\theta\)-methods for damped stochastic oscillators.https://www.zbmath.org/1453.650152021-02-27T13:50:00+00:00"Citro, Vincenzo"https://www.zbmath.org/authors/?q=ai:citro.vincenzo"D'Ambrosio, Raffaele"https://www.zbmath.org/authors/?q=ai:dambrosio.raffaeleSummary: We analyze long-term properties of stochastic \(\theta\)-methods for damped linear stochastic oscillators. The presented a-priori analysis of the error in the correlation matrix allows to infer the long-time behaviour of stochastic \(\theta\)-methods and their capability to reproduce the same long-term features of the continuous dynamics. The theoretical analysis is also supported by a selection of numerical experiments.Solving electrical impedance tomography with deep learning.https://www.zbmath.org/1453.650412021-02-27T13:50:00+00:00"Fan, Yuwei"https://www.zbmath.org/authors/?q=ai:fan.yuwei"Ying, Lexing"https://www.zbmath.org/authors/?q=ai:ying.lexingSummary: This paper introduces a new approach for solving electrical impedance tomography (EIT) problems using deep neural networks. The mathematical problem of EIT is to invert the electrical conductivity from the Dirichlet-to-Neumann (DtN) map. Both the forward map from the electrical conductivity to the DtN map and the inverse map are high-dimensional and nonlinear. Motivated by the linear perturbative analysis of the forward map and based on a numerically low-rank property, we propose compact neural network architectures for the forward and inverse maps for both 2D and 3D problems. Numerical results demonstrate the efficiency of the proposed neural networks.Random fuzzy numbers generation with cubic Hermit membership function and its application in simulation.https://www.zbmath.org/1453.650132021-02-27T13:50:00+00:00"Fathi-Vajargah, Behrouz"https://www.zbmath.org/authors/?q=ai:fathi-vajargah.behrouz"Ghasemalipour, Sara"https://www.zbmath.org/authors/?q=ai:ghasemalipour.saraSummary: Fuzzy numbers play a very important role in linguistic decision making and some other fuzzy application systems. Several strategies have been proposed for generating fuzzy numbers. In this paper, we introduce a new approach for generating fuzzy numbers based on cubic Hermit curves. Fuzzy numbers with polynomial membership function are not often in \((0,1)\) and we use from cubic Hermit curves for generating fuzzy numbers. Fuzzy numbers with cubic Hermit membership function are always within the \((0,1)\). Finally, we apply these numbers in simulation.Upwinding and artificial viscosity for robust discontinuous Galerkin schemes of two-phase flow in mass conservation form.https://www.zbmath.org/1453.760722021-02-27T13:50:00+00:00"Jayasinghe, Savithru"https://www.zbmath.org/authors/?q=ai:jayasinghe.savithru"Darmofal, David L."https://www.zbmath.org/authors/?q=ai:darmofal.david-l"Allmaras, Steven R."https://www.zbmath.org/authors/?q=ai:allmaras.steven-r"Dow, Eric"https://www.zbmath.org/authors/?q=ai:dow.eric"Galbraith, Marshall C."https://www.zbmath.org/authors/?q=ai:galbraith.marshall-cSummary: High-order discretizations have become increasingly popular across a wide range of applications, including reservoir simulation. However, the lack of stability and robustness of these discretizations for advection-dominant problems prevent them from being widely adopted. This paper presents work towards improving the stability and robustness of the discontinuous Galerkin (DG) finite element scheme, for advection-dominant two-phase flow problems in particular. A linearized analysis of the two-phase flow equations is used to show that a standard DG discretization of the two-phase flow equations in mass conservation form results in a neutrally stable semi-discrete system in the advection-dominant limit. Furthermore, the analysis is also used to propose additional terms to the DG method which linearly stabilize the discretization. These additional terms are derived by comparing the linearized equations in mass conservation form against an upwinded pressure-saturation form of the equations. Next, a partial differential equation-based artificial viscosity method is proposed for the Buckley-Leverett and two-phase flow equations, as a means of mitigating Gibbs oscillations in high-order discretizations and ensuring convergence to physical solutions. The modified DG method with artificial viscosity is demonstrated on a two-phase flow problem with heterogeneous rock permeabilities, where the high-order discretizations significantly outperform a conventional first-order approach in terms of computational cost required to achieve a given level of error in an output of interest.Efficient solution techniques for two-phase flow in heterogeneous porous media using exact Jacobians.https://www.zbmath.org/1453.650632021-02-27T13:50:00+00:00"Büsing, Henrik"https://www.zbmath.org/authors/?q=ai:busing.henrikSummary: Two efficient and scalable numerical solution methods will be compared using exact Jacobians to solve the fully coupled Newton systems arising during fully implicit discretization of the equations for two-phase flow in porous media. These methods use algebraic multigrid (AMG) to solve the linear systems in every Newton step. The algebraic multigrid methods rely on (i) a Schur Complement Reduction (SCR-AMG) and (ii) a Constrained Pressure Residual method (CPR-AMG) to decouple elliptic and hyperbolic contributions. Both methods employ automatic differentiation (AD) to calculate exact Jacobians and are compared with finite difference (FD) approximations of the Jacobian. The superiority of AD is shown by several numerical test cases from the field of \(\mathrm{CO_2}\) geo-sequestration comprising two- and three-dimensional examples. A weak scaling test on JUQUEEN, a BlueGene/Q supercomputer, demonstrates the efficiency and scalability of both methods. To achieve maximum comparability and reproducibility, the Portable Extensible Toolkit for Scientific Computation (PETSc) is used as framework for the comparison of all solvers.Quasi-implicit treatment of velocity-dependent mobilities in underground porous media flow simulation.https://www.zbmath.org/1453.761052021-02-27T13:50:00+00:00"Patacchini, Leonardo"https://www.zbmath.org/authors/?q=ai:patacchini.leonardo"de Loubens, Romain"https://www.zbmath.org/authors/?q=ai:de-loubens.romainSummary: Quasi-implicit schemes for treating velocity-dependent mobilities in underground porous media flow simulation, occurring when modeling non-Newtonian and non-Darcy effects as well as capillary desaturation, are presented. With low-order finite-volume discretizations, the principle is to evaluate mobilities at cell edges using normal velocity components calculated implicitly, and transverse velocity components calculated explicitly (i.e., based on the previously converged time-step); the pressure gradient driving the flow is, as usual, treated implicitly. On 3D hexahedral meshes, the proposed schemes require the same 7-point stencil as that of common semi-implicit schemes where mobilities are evaluated with an entirely explicit velocity argument. When formulated appropriately, their higher level of implicitness however places them, in terms of numerical stability, closer to ``real'' fully implicit schemes requiring at least a 19-point stencil. A von Neumann stability analysis of these proposed schemes is performed on a simplified pressure equation, representative of both single-phase and multiphase flows, following an approach previously used by the authors to study semi-implicit schemes. Whereas the latter are subject to stability constraints which limit their usage in certain cases where the logarithmic derivative of mobility with respect to velocity is large in magnitude, the former are unconditionally stable for 1D and 2D flows, and only subject to weak restrictions for 3D flows.An optimal sixteenth order family of methods for solving nonlinear equations and their basins of attraction.https://www.zbmath.org/1453.650982021-02-27T13:50:00+00:00"Ćebić, Dejan"https://www.zbmath.org/authors/?q=ai:cebic.dejan"Ralević, Nebojša M. Ralević"https://www.zbmath.org/authors/?q=ai:ralevic.nebojsa-m-ralevic"Marčeta, Marina"https://www.zbmath.org/authors/?q=ai:marceta.marinaSummary: We propose a new family of iterative methods for finding the simple roots of nonlinear equation. The proposed method is four-point method with convergence order 16, which consists of four steps: the Newton step, an optional fourth order iteration scheme, an optional eighth order iteration scheme and the step constructed using the divided difference. By reason of the new iteration scheme requiring four function evaluations and one first derivative evaluation per iteration, the method satisfies the optimality criterion in the sense of Kung-Traub's conjecture and achieves a high efficiency index \(16^{1/5} \approx 1.7411\). Computational results support theoretical analysis and confirm the efficiency. The basins of attraction of the new presented algorithms are also compared to the existing methods with encouraging results.Finite-volume simulation of capillary-dominated flow in matrix-fracture systems using interface conditions.https://www.zbmath.org/1453.860022021-02-27T13:50:00+00:00"Alali, Ammar H."https://www.zbmath.org/authors/?q=ai:alali.ammar-h"Hamon, François P."https://www.zbmath.org/authors/?q=ai:hamon.francois-p"Mallison, Bradley T."https://www.zbmath.org/authors/?q=ai:mallison.bradley-t"Tchelepi, Hamdi A."https://www.zbmath.org/authors/?q=ai:tchelepi.hamdi-aSummary: In numerical simulations of multiphase flow and transport in fractured porous media, the estimation of the hydrocarbon recovery requires accurately predicting the capillary-driven imbibition rate of the wetting phase initially present in the fracture into the low-permeability matrix. In the fully implicit finite-volume scheme, this entails a robust methodology that captures the capillary flux at the interface between the matrix and the fracture even when very coarse cells are used to discretize the matrix. Here, we investigate the application of discrete interface conditions at the matrix-fracture interface to improve the accuracy of the flux computation without relying on extreme grid refinement. In particular, we study the interaction of the upwinding scheme with the discrete interface conditions. Considering first capillary-dominated spontaneous imbibition and then forced imbibition with viscous, buoyancy, and capillary forces, we illustrate the importance of the interface conditions to accurately capture the matrix-fracture flux and correctly represent the flow dynamics in the problem.An efficient parallel algorithm for 3D magnetotelluric modeling with edge-based finite element.https://www.zbmath.org/1453.860122021-02-27T13:50:00+00:00"Zhu, Xiaoxiong"https://www.zbmath.org/authors/?q=ai:zhu.xiaoxiong"Liu, Jie"https://www.zbmath.org/authors/?q=ai:liu.jie.5|liu.jie.2|liu.jie.3|liu.jie|liu.jie.6|liu.jie.1|liu.jie.4|liu.jie.7"Cui, Yian"https://www.zbmath.org/authors/?q=ai:cui.yian"Gong, Chunye"https://www.zbmath.org/authors/?q=ai:gong.chunyeSummary: Three-dimensional magnetotelluric modeling algorithm of high accuracy and high efficiency is required for data interpretation and inversion. In this paper, edge-based finite element method with unstructured mesh is used to solve 3D magnetotelluric problem. Two boundary conditions -- Dirichlet boundary condition and Neumann boundary condition -- are set for cross-validation and comparison. We propose an efficient parallel algorithm to speed up computation and improve efficiency. The algorithm is based on distributed matrix storage and has three levels of parallelism. The first two are process level parallelization for frequencies and matrix solving, and the last is thread-level parallelization for loop unrolling. The algorithm is validated by several model studies. Scalability tests have been performed on two distributed-memory HPC platforms, one consists of Intel Xeon E5-2660 microprocessors and the other consists of Phytium FT2000 Plus microprocessors. On Intel platform, computation time of our algorithm solving Dublin Test Model-1 with 3,756,373 edges at 21 frequencies is 365 s on 2520 cores. The speedup and efficiency are 1609 and 60\% compared to 100 cores. On Phytium platform, scalability test shows that the speedup from 256 cores to 86,016 cores has been increased to 11,255.A fast algorithm for radiative transport in isotropic media.https://www.zbmath.org/1453.653622021-02-27T13:50:00+00:00"Ren, Kui"https://www.zbmath.org/authors/?q=ai:ren.kui"Zhang, Rongting"https://www.zbmath.org/authors/?q=ai:zhang.rongting"Zhong, Yimin"https://www.zbmath.org/authors/?q=ai:zhong.yiminSummary: Constructing efficient numerical solution methods for the equation of radiative transfer (ERT) remains as a challenging task in scientific computing despite of the tremendous development on the subject in recent years. We present in this work a simple fast computational algorithm for solving the ERT in isotropic media. The algorithm we developed has two steps. In the first step, we solve a volume integral equation for the angularly-averaged ERT solution using iterative schemes such as the GMRES method. The computation in this step is accelerated with a fast multipole method (FMM). In the second step, we solve a scattering-free transport equation to recover the angular dependence of the ERT solution. The algorithm does not require the underlying medium be homogeneous. We present numerical simulations under various scenarios to demonstrate the performance of the proposed numerical algorithm for both homogeneous and heterogeneous media.A unified approach for deriving optimal finite differences.https://www.zbmath.org/1453.652252021-02-27T13:50:00+00:00"Kumari, Komal"https://www.zbmath.org/authors/?q=ai:kumari.komal"Bhattacharya, Raktim"https://www.zbmath.org/authors/?q=ai:bhattacharya.raktim"Donzis, Diego A."https://www.zbmath.org/authors/?q=ai:donzis.diego-aSummary: A unified approach to derive optimal finite differences is presented which combines three critical elements for numerical performance especially for multi-scale physical problems, namely, order of accuracy, spectral resolution and stability. The resulting mathematical framework reduces to a minimization problem subjected to equality and inequality constraints. We show that the framework can provide analytical results for optimal schemes and their numerical performance including, for example, the type of errors that appear for spectrally optimal schemes. By coupling the problem in this unified framework, one can effectively decouple the requirements for order of accuracy and spectral resolution, for example. Alternatively, we show how the framework exposes the tradeoffs between e.g. accuracy and stability and how this can be used to construct explicit schemes that remain stable with very large time steps. We also show how spectrally optimal schemes only bias odd-order derivatives to remain stable, at the expense of accuracy, while leaving even-order derivatives with symmetric coefficients. Schemes constructed within this framework are tested for diverse model problems with an emphasis on reproducing the physics accurately.A volume of fluid framework for interface-resolved simulations of vaporizing liquid-gas flows.https://www.zbmath.org/1453.762152021-02-27T13:50:00+00:00"Palmore, John"https://www.zbmath.org/authors/?q=ai:palmore.john"Desjardins, Olivier"https://www.zbmath.org/authors/?q=ai:desjardins.olivierSummary: This work demonstrates a computational framework for simulating vaporizing, liquid-gas flows. It is developed for the general vaporization problem [\textit{L. Rueda Villegas} et al., ibid. 316, 789--813 (2016; Zbl 1349.76382)] which solves the vaporization rate based as from the local thermodynamic equilibrium of the liquid-gas system. This includes the commonly studied vaporization regimes of film boiling and isothermal evaporation. The framework is built upon a Cartesian grid solver for low-Mach, turbulent flows [the second author et al., ibid. 227, No. 15, 7125--7159 (2008; Zbl 1201.76139)] which has been modified to handle multiphase flows with large density ratios [the second author et al., ``Direct numerical and large-eddy simulation of primary atomization in complex geometries'', At. Sprays 23, No. 11, 1001--1048 (2013; \url{doi:10.1615/AtomizSpr.2013007679})]. Interface transport is performed using an unsplit volume of fluid solver [\textit{M. Owkes} and the second author, J. Comput. Phys. 270, 587--612 (2014; Zbl 1349.76636)]. A novel, divergence-free extrapolation technique is used to create a velocity field that is suitable for interface transport. Sharp treatments are used for the vapor mass fractions and temperature fields. The pressure Poisson equation is treated using the Ghost Fluid Method. Interface equilibrium at the interface is computed using the Clausius-Clapeyron relation, and is coupled to the flow solver using a monotone, unconditionally stable scheme. It will be shown that correct prediction of the interface properties is fundamental to accurate simulations of the vaporization process. The convergence and accuracy of the proposed numerical framework is verified against solutions in one, two, and three dimensions. The simulations recover first order convergence under temporal and spatial refinement for the general vaporization problem. The work is concluded with a demonstration of unsteady vaporization of a droplet at intermediate Reynolds number.On the centroidal mean Newton's method for simple and multiple roots of nonlinear equations.https://www.zbmath.org/1453.651032021-02-27T13:50:00+00:00"Verma, K. L."https://www.zbmath.org/authors/?q=ai:verma.k-lSummary: In this paper, the convergence behaviour of a variant of Newton's method based on the centroidal mean is considered. The convergence properties of this method for solving equations which have simple or multiple roots have been discussed. It is shown that it converges cubically to simple roots with efficiency index is 1.442 and linearly to multiple roots. Moreover, the values of the corresponding asymptotic error constants of convergence are determined. Theoretical results have been verified on the relevant numerical problems. The proposed new method has the advantage of evaluating only the first derivative and less number of iterations to achieve third order accuracy. A comparison of the efficiency of this method with other mean-based Newton's methods, based on the arithmetic, geometric and harmonic means, is also included. Convergences to the root and error propagation with iteration are exhibited graphically with iterations.B-spline collocation method for boundary value problems in complex domains.https://www.zbmath.org/1453.654212021-02-27T13:50:00+00:00"Hidayat, Mas Irfan P."https://www.zbmath.org/authors/?q=ai:hidayat.mas-irfan-purbawanto"Ariwahjoedi, Bambang"https://www.zbmath.org/authors/?q=ai:ariwahjoedi.bambang"Parman, Setyamartana"https://www.zbmath.org/authors/?q=ai:parman.setyamartanaSummary: In this paper, an over-determined, global collocation method based upon B-spline basis functions is presented for solving boundary value problems in complex domains. The method was truly meshless approach, hence simple and efficient to programme. In the method, any governing equations were discretised by global B-spline approximation as the B-spline interpolants. As the interpolating B-spline basis functions were chosen, the present method also posed the Kronecker delta property allowing boundary conditions to be incorporated efficiently. The present method showed high accuracy for elliptic partial differential equations in arbitrary domain with Neumann boundary conditions. For coupled Poisson problems with complex Neumann boundary conditions, the boundary collocation approach was adopted and applied in a simple and less costly manner to further improve the accuracy and stability. Applications from elasticity problems were given to demonstrate the efficacy and capability of the present method. In addition, the relation between accuracy and stability for the method was better justified by the new effective condition number given in literature.Method of fundamental solutions without fictitious boundary for anisotropic elasticity problems based on mechanical equilibrium desingularization.https://www.zbmath.org/1453.654332021-02-27T13:50:00+00:00"Liu, Qingguo"https://www.zbmath.org/authors/?q=ai:liu.qingguo"Šarler, Božidar"https://www.zbmath.org/authors/?q=ai:sarler.bozidarSummary: In this chapter, an Improved Non-singular Method of Fundamental Solutions (INMFS) is developed for solving the 2D anisotropic linear elasticity problems. In the INMFS, the artificial boundary, present in the classical Method of Fundamental Solutions (MFS), is removed. The singularities are substituted by the normalized area integrals of the Fundamental Solution (FS) over small squares, covering the source points that intersect with the collocation points. The singularities of the fundamental traction are dealt with considering the mechanical equilibrium, calculated by the boundary integration of the forces on the considered body. This more appropriate approach avoids solving the problem two times as in Non-singular MFS (NMFS), developed by \textit{Q. G. Liu} and \textit{B. Šarler} in [Eng. Anal. Bound. Elem. 45, 68--78 (2014; Zbl 1297.74019)]. The integral over a small disk in NMFS is replaced by a small square in INMFS, amenable to be solved analytically. The viability and superiority of INMFS in comparison with the MFS and the NMFS is assessed in details. The advantage of having no artificial boundary and the straightforward implementation of the INMFS on problems with different materials in contact is demonstrated.
For the entire collection see [Zbl 1445.65001].Application of quadtrees in the method of fundamental solutions using multi-level tools.https://www.zbmath.org/1453.654322021-02-27T13:50:00+00:00"Gáspár, Csaba"https://www.zbmath.org/authors/?q=ai:gaspar.csabaSummary: The traditional version of the Method of Fundamental Solutions is revisited, which is based on using external sources. The sources are defined in a completely automatic way using a quadtree cell system controlled by the boundary of the domain. This results in a spatial density distribution of sources which decreases rapidly when going far away from the boundary. A similar technique is also proposed, when the sources are located along the boundary (which can be automated easily) and the collocation points are moved into the interior of the domain. In this case, the boundary conditions have to be properly redefined at the inner collocation points, This is done by using boundary-controlled quadtrees again. Both techniques can be embedded in a multi-level context in a natural way. The accuracy of the resulting methods are less than that of the traditional Method of Fundamental Solutions, but it is still acceptable. However, the computational cost is more moderate and the problems of singularity and the extremely ill-conditioned matrices are avoided.
For the entire collection see [Zbl 1445.65001].Automatic differentiation using operator overloading (ADOO) for implicit resolution of hyperbolic single phase and two-phase flow models.https://www.zbmath.org/1453.650492021-02-27T13:50:00+00:00"Fraysse, François"https://www.zbmath.org/authors/?q=ai:fraysse.francois"Saurel, Richard"https://www.zbmath.org/authors/?q=ai:saurel.richardSummary: Implicit time integration schemes are widely used in computational fluid dynamics to speed-up computations. Indeed, implicit schemes usually allow for less stringent time-step stability constraints than their explicit counterpart. The derivation of an implicit scheme is however a challenging and time-consuming task, increasing substantially with the model equations complexity since this method usually requires fairly accurate evaluation of the spatial scheme's matrix Jacobian. This article presents a flexible method to overcome the difficulties associated to the computation of the derivatives, based on the forward mode of automatic differentiation using operator overloading (ADOO). Flexibility and simplicity of the method are illustrated through implicit resolution of various flow models of increasing complexity such as the compressible Euler equations, a two-phase flow model in full equilibrium [\textit{S. Le Martelot} et al., ``Towards the direct numerical simulation of nucleate boiling flows'', Int. J. Multiphase Flow 66, 62--78 (2014; \url{doi:10.1016/j.ijmultiphaseflow.2014.06.010})] and a symmetric variant [the second author et al., J. Fluid Mech. 495, 283--321 (2003; Zbl 1080.76062)] of the two-phase flow model of \textit{M. R. Baer} and \textit{J. W. Nunziato} [Int. J. Multiphase Flow 12, 861--889 (1986; Zbl 0609.76114)] dealing with mixtures in total disequilibrium.Higher-order optimized hybrid Robert-Asselin type time filters.https://www.zbmath.org/1453.652852021-02-27T13:50:00+00:00"Maurya, Praveen K."https://www.zbmath.org/authors/?q=ai:maurya.praveen-k"Rajpoot, Manoj K."https://www.zbmath.org/authors/?q=ai:rajpoot.manoj-k"Yadav, Vivek S."https://www.zbmath.org/authors/?q=ai:yadav.vivek-sSummary: In this paper, a class of optimized filters based on hybrid blending of implicit and explicit filters are formulated for the three time-level leapfrog time integration scheme. As the leapfrog method is the most recurrently employed in ocean and atmospheric simulations, however, it also admits a spurious mode in the numerical computations. To subdue the computational mode, the developed hybrid time filters are also optimized using constraints arising from error dynamics. The optimized filters are adept in suppressing the computational mode(s) more efficaciously without altering the physical mode. Finally, the developed hybrid filters coupled with centered and compact spatial discretization schemes are also authenticated for the numerical solutions to one-dimensional (1D) convection equation, two-dimensional (2D) dispersive rotating shallow water equation, and unsteady 2D incompressible Navier-Stokes equation at different Reynolds numbers. Present solutions are also compared with results reported in the literature.A high-order conservative remap for discontinuous Galerkin schemes on curvilinear polygonal meshes.https://www.zbmath.org/1453.653332021-02-27T13:50:00+00:00"Lipnikov, Konstantin"https://www.zbmath.org/authors/?q=ai:lipnikov.konstantin-n"Morgan, Nathaniel"https://www.zbmath.org/authors/?q=ai:morgan.nathaniel-rSummary: A data transfer (called later remap) of physical fields between two meshes is an important step of arbitrary Lagrangian-Eulerian (ALE) simulations. This step is challenging for high-order discontinuous Galerkin schemes since the Lagrangian flow motion leads to high-order meshes with curved faces. It becomes even more challenging for unstructured polygonal meshes that do not have a polynomial map from the reference to a current cell. We propose and analyze a new framework to create remap schemes on curvilinear polygonal meshes based on the theory of virtual element projectors. We derive a conservative remap scheme that is high-order accurate in space and time. The properties of this scheme are studied numerically for smooth and discontinuous fields on unstructured quadrilateral and polygonal meshes.Simulating from irregular data: kernel Carlo simulation.https://www.zbmath.org/1453.621142021-02-27T13:50:00+00:00"Howe, J. Andrew"https://www.zbmath.org/authors/?q=ai:howe.j-andrewSummary: In this paper, we address the problem of simulating from a data-generating process for which the observed data do not follow a regular probability distribution. One existing method for doing this is bootstrapping, but it is incapable of interpolating between observed data. For univariate or bivariate data, in which a mixture structure can easily be identified, we could instead simulate from a Gaussian mixture model. In general, though, we would have the problem of identifying and estimating the mixture model. Instead of these, we introduce a non-parametric method for simulating datasets like this: Kernel Carlo Simulation. Our algorithm begins by using kernel density estimation to build a target probability distribution. Then, an envelope function that is guaranteed to be higher than the target distribution is created. We then use simple accept-reject sampling. Our approach is more flexible than others, can simulate intelligently across gaps in the data, and requires no subjective modelling decisions. With several univariate and multivariate examples, we show that our method returns simulated datasets that, compared with the observed data, retain the covariance structures and have distributional characteristics that are remarkably similar.Discontinuous Galerkin methods for short pulse type equations via hodograph transformations.https://www.zbmath.org/1453.653492021-02-27T13:50:00+00:00"Zhang, Qian"https://www.zbmath.org/authors/?q=ai:zhang.qian"Xia, Yinhua"https://www.zbmath.org/authors/?q=ai:xia.yinhuaSummary: In the present paper, we consider the discontinuous Galerkin (DG) methods for solving short pulse (SP) type equations. The short pulse equation has been shown to be completely integrable, which admits the loop-soliton, cuspon-soliton solutions as well as smooth-soliton solutions. Through hodograph transformations, these nonclassical solutions can be profiled as the smooth solutions of the coupled dispersionless (CD) system or the sine-Gordon equation. Therefore, DG methods can be developed for the CD system or the sine-Gordon equation to simulate the loop-soliton or cuspon-soliton solutions of the SP equation. The conservativeness or dissipation of the Hamiltonian or momentum for the semi-discrete DG schemes can be proved. Also we modify the above DG schemes and obtain an integration DG scheme. Theoretically the a-priori error estimates have been provided for the momentum conserved DG scheme and the integration DG scheme. We also propose the DG scheme and the integration DG scheme for the sine-Gordon equation, in case the SP equation can not be transformed to the CD system. All these DG schemes can be applied to the generalized or modified SP type equations. Numerical experiments are provided to illustrate the optimal order of accuracy and capability of these DG schemes.A nonlinear elimination preconditioned inexact Newton method for blood flow problems in human artery with stenosis.https://www.zbmath.org/1453.651092021-02-27T13:50:00+00:00"Luo, Li"https://www.zbmath.org/authors/?q=ai:luo.li.1"Shiu, Wen-Shin"https://www.zbmath.org/authors/?q=ai:shiu.wen-shin"Chen, Rongliang"https://www.zbmath.org/authors/?q=ai:chen.rongliang"Cai, Xiao-Chuan"https://www.zbmath.org/authors/?q=ai:cai.xiao-chuanSummary: Simulation of blood flows in the human artery is a promising tool for understanding the hemodynamics. The blood flow is often smooth in a healthy artery, but may become locally chaotic in a diseased artery with stenosis, and as a result, a traditional solver may take many iterations to converge or does not converge at all. To overcome the problem, we develop a nonlinearly preconditioned Newton method in which the variables associated with the stenosis are iteratively eliminated and then a global Newton method is applied to the smooth part of the system. More specifically, we model the blood flow in a patient-specific artery based on the unsteady incompressible Navier-Stokes equations with resistive boundary conditions discretized by a fully implicit finite element method. The resulting nonlinear system at each time step is solved by using an inexact Newton method with a domain decomposition based Jacobian solver. To improve the convergence and robustness of the Newton method for arteries with stenosis, we develop an adaptive restricted region-based nonlinear elimination preconditioner which performs subspace correction to remove the local high nonlinearities. Numerical experiments for several cerebral arteries are presented to demonstrate the superiority of the proposed algorithm over the classical method with respect to some physical and numerical parameters. We also report the parallel scalability of the proposed algorithm on a supercomputer with thousands of processor cores.A fast spectral method for the Uehling-Uhlenbeck equation for quantum gas mixtures: homogeneous relaxation and transport coefficients.https://www.zbmath.org/1453.653682021-02-27T13:50:00+00:00"Wu, Lei"https://www.zbmath.org/authors/?q=ai:wu.lei.3|wu.lei.1|wu.lei|wu.lei.2|wu.lei.4Summary: A fast spectral method (FSM) is developed to solve the Uehling-Uhlenbeck equation for quantum gas mixtures with generalized differential cross-sections. The computational cost of the proposed FSM is \(O(M^{d_v - 1} N^{d_v + 1} \log N)\), where \(d_v\) is the dimension of the problem, \(M^{d_v - 1}\) is the number of discrete solid angles, and \(N\) is the number of frequency nodes in each direction. Spatially-homogeneous relaxation problems are used to demonstrate that the FSM conserves mass and momentum/energy to the machine and spectral accuracy, respectively. Based on the variational principle, transport coefficients such as the shear viscosity, thermal conductivity, and diffusion are calculated by the FSM, which agree well with the analytical solutions. Then, the FSM is applied to find the accurate transport coefficients through an iterative scheme for the linearized quantum Boltzmann equation. The shear viscosity and thermal conductivity of three-dimensional quantum Fermi and Bose gases interacting through hard-sphere potential are calculated. For Fermi gas, the relative difference between the accurate and variational transport coefficients increases with fugacity; for Bose gas, the relative difference in thermal conductivity has similar behavior as the gas moves from the classical to degenerate limits, but the relative difference in shear viscosity decreases when the fugacity increases. Finally, the viscosity and diffusion coefficients are calculated for a two-dimensional equal-mole mixture of Fermi gases. When the molecular masses of the two components are the same, our numerical results agree with the variational solutions. However, when the molecular mass ratio is not one, large discrepancies between the accurate and variational results are observed; our results are reliable because (i) the method does not rely on any assumption on the form of velocity distribution function and (ii) the ratio between shear viscosity and entropy density satisfies the minimum bound predicted by the string theory.Data-driven model reduction for a class of semi-explicit DAEs using the Loewner framework.https://www.zbmath.org/1453.651992021-02-27T13:50:00+00:00"Antoulas, Athanasios C."https://www.zbmath.org/authors/?q=ai:antoulas.athanasios-c"Gosea, Ion Victor"https://www.zbmath.org/authors/?q=ai:gosea.ion-victor"Heinkenschloss, Matthias"https://www.zbmath.org/authors/?q=ai:heinkenschloss.matthiasSummary: This paper introduces a modified version of the recently proposed data-driven Loewner framework to compute reduced order models (ROMs) for a class of semi-explicit differential algebraic equation (DAE) systems, which include the semi-discretized linearized Navier-Stokes/Oseen equations. The modified version estimates the polynomial part of the original transfer function from data and incorporates this estimate into the Loewner ROM construction. Without this proposed modification the transfer function of the Loewner ROM is strictly proper, i.e., goes to zero as the magnitude of the frequency goes to infinity, and therefore may have a different behavior for large frequencies than the transfer function of the original system. The modification leads to a Loewner ROM with a transfer function that has a strictly proper and a polynomial part, just as the original model. This leads to better approximations for transfer function components in which the coefficients in the polynomial part are not too small. The construction of the improved Loewner ROM is described and the improvement is demonstrated on a large-scale system governed by the semi-discretized Oseen equations.
For the entire collection see [Zbl 1445.34004].Improvement of Rosenbrock-Wanner method RODASP. Enhanced coefficient set and MATLAB implementation.https://www.zbmath.org/1453.652032021-02-27T13:50:00+00:00"Steinebach, Gerd"https://www.zbmath.org/authors/?q=ai:steinebach.gerdSummary: Rosenbrock-Wanner methods for solving index-one DAEs usually suffer from order reduction to order \(p = 1\) when the Jacobian matrix is not exactly computed in every time step. This may even happen when the Jacobian matrix is updated in every step, but numerically evaluated by finite differences. Recently, \textit{T. Jax} [A rooted-tree based derivation of ROW-type methods with arbitrary Jacobian entries for solving index-one DAEs. Dissertation, Bergische Universität Wuppertal (2019)] could derive new order conditions for the avoidance of such order reduction phenomena. In this paper we present an improvement of the known Rosenbrock-Wanner method \texttt{rodasp} [\textit{G. Steinebach}, Order-reduction of ROW-methods for DAEs and method of lines applications, Preprint-Nr. 1741. FB Mathematik, TH Darmstadt (1995)]. It is possible to modify its coefficient set such that only an order reduction to \(p = 2\) occurs. Several numerical tests indicate that the new method is more efficient than \texttt{rodasp} and the original method \texttt{rodas} from \textit{E. Hairer} and \textit{G. Wanner} [Solving ordinary differential equations. II: Stiff and differential-algebraic problems. 2nd rev. ed. Berlin: Springer (1996; Zbl 0859.65067)]. When additionally measures for the efficient evaluation of the Jacobian matrix are applied, the method can compete with the standard integrator \texttt{ode15s} of MATLAB.
For the entire collection see [Zbl 1445.34004].Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes.https://www.zbmath.org/1453.652432021-02-27T13:50:00+00:00"Arbogast, Todd"https://www.zbmath.org/authors/?q=ai:arbogast.todd"Huang, Chieh-Sen"https://www.zbmath.org/authors/?q=ai:huang.chieh-sen"Zhao, Xikai"https://www.zbmath.org/authors/?q=ai:zhao.xikaiSummary: We consider numerical approximation of the degenerate advection-diffusion equation, which is formally parabolic but may exhibit hyperbolic behavior. We develop both explicit and implicit finite volume weighted essentially non-oscillatory (WENO) schemes in multiple space dimensions on non-uniform computational meshes. The diffusion degeneracy is reformulated through the use of the Kirchhoff transformation. Space is discretized using WENO reconstructions with adaptive order (WENO-AO), which have several advantages, including the avoidance of negative linear weights and the ability to handle irregular computational meshes. A special two-stage WENO reconstruction procedure is developed to handle degenerate diffusion. Element averages of the solution are first reconstructed to give point values of the solution, and these point values are in turn used to reconstruct the Kirchhoff transform variable of the diffusive flux. Time is discretized using the method of lines and a Runge-Kutta time integrator. We use Strong Stability Preserving (SSP) Runge-Kutta methods for the explicit schemes, which have a severe parabolically scaled time step restriction to maintain stability. We also develop implicit Runge-Kutta methods. SSP methods are only conditionally stable, so we discuss the use of L-stable Runge-Kutta methods. We present in detail schemes that are third order in both space and time in one and two space dimensions using non-uniform meshes of intervals or quadrilaterals. Efficient implementation is described for computational meshes that are logically rectangular. Through a von Neumann (or Fourier mode) stability analysis, we show that smooth solutions to the linear problem are unconditionally L-stable on uniform computational meshes when using an implicit Radau IIA Runge-Kutta method. Computational results show the ability of the schemes to accurately approximate challenging test problems.Least-squares collocation for higher-index DAEs: global approach and attempts toward a time-stepping version.https://www.zbmath.org/1453.652022021-02-27T13:50:00+00:00"Hanke, Michael"https://www.zbmath.org/authors/?q=ai:hanke.michael"März, Roswitha"https://www.zbmath.org/authors/?q=ai:marz.roswithaSummary: Overdetermined polynomial least-squares collocation for two-point boundary value problems for higher index differential-algebraic equations shows excellent convergence properties while at the same time being only slightly more expensive than the widely used collocation method for ordinary differential equations by piecewise polynomials. In the present paper, basic properties of this method when applied to initial value problems by a windowing technique are proven. Some examples are provided in order to show the potential of time-stepping approach.
For the entire collection see [Zbl 1445.34004].Inter/extrapolation-based multirate schemes: a dynamic-iteration perspective.https://www.zbmath.org/1453.652002021-02-27T13:50:00+00:00"Bartel, Andreas"https://www.zbmath.org/authors/?q=ai:bartel.andreas"Günther, Michael"https://www.zbmath.org/authors/?q=ai:gunther.michaelSummary: Multirate behavior of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) is characterized by widely separated time constants in different components of the solution or different additive terms of the right-hand side. Here, classical multirate schemes are dedicated solvers, which apply (e.g.) micro and macro steps to resolve fast and slow changes in a transient simulation accordingly. The use of extrapolation and interpolation procedures is a genuine way for coupling the different parts, which are defined on different time grids.
This paper contains for the first time, to the best knowledge of the authors, a complete convergence theory for inter/extrapolation-based multirate schemes for both ODEs and DAEs of index one, which are based on the fully-decoupled approach, the slowest-first and the fastest-first approach. The convergence theory is based on linking these schemes to multirate dynamic iteration schemes, i.e., dynamic iteration schemes without further iterations. This link defines naturally stability conditions for the DAE case.
For the entire collection see [Zbl 1445.34004].A high resolution PDE approach to quadrilateral mesh generation.https://www.zbmath.org/1453.650452021-02-27T13:50:00+00:00"Marcon, Julian"https://www.zbmath.org/authors/?q=ai:marcon.julian"Kopriva, David A."https://www.zbmath.org/authors/?q=ai:kopriva.david-a"Sherwin, Spencer J."https://www.zbmath.org/authors/?q=ai:sherwin.spencer-j"Peiró, Joaquim"https://www.zbmath.org/authors/?q=ai:peiro.joaquimSummary: We describe a high order technique to generate quadrilateral decompositions and meshes for complex two dimensional domains using spectral elements in a field guided procedure. Inspired by cross field methods, we never actually compute crosses. Instead, we compute a high order accurate guiding field using a continuous Galerkin (CG) or discontinuous Galerkin (DG) spectral element method to solve a Laplace equation for each of the field variables using the open source code \textit{Nektar++}. The spectral method provides spectral convergence and sub-element resolution of the fields. The DG approximation allows meshing of corners that are not multiples of \(\pi / 2\) in a discretization consistent manner, when needed. The high order field can then be exploited to accurately find irregular nodes, and can be accurately integrated using a high order separatrix integration method to avoid features like limit cycles. The result is a mesh with naturally curved quadrilateral elements that do not need to be curved \textit{a posteriori} to eliminate invalid elements. The mesh generation procedure is implemented in the open source mesh generation program \textit{NekMesh}.Robust explicit relaxation technique for solving the Green-Naghdi equations.https://www.zbmath.org/1453.653212021-02-27T13:50:00+00:00"Guermond, Jean-Luc"https://www.zbmath.org/authors/?q=ai:guermond.jean-luc"Popov, Bojan"https://www.zbmath.org/authors/?q=ai:popov.boyan"Tovar, Eric"https://www.zbmath.org/authors/?q=ai:tovar.eric"Kees, Chris"https://www.zbmath.org/authors/?q=ai:kees.chrisSummary: This paper revisits an original relaxation technique introduced in [\textit{N. Favrie} and \textit{S. Gavrilyuk}, Nonlinearity 30, No. 7, 2718--2736 (2017; Zbl 1432.65120)] for solving the Green-Naghdi equations. We propose a version of that method and a space/time approximation thereof that is scale invariant. The approximation in time is explicit and the approximation in space uses a length scale for the relaxation that is proportional to the mesh size. The new method is compatible with dry states and is provably positivity preserving under the appropriate CFL condition. The method is numerically validated against manufactured solutions and is illustrated by comparison with experimental results.Locally adaptive pseudo-time stepping for high-order flux reconstruction.https://www.zbmath.org/1453.653352021-02-27T13:50:00+00:00"Loppi, N. A."https://www.zbmath.org/authors/?q=ai:loppi.niki-a"Witherden, F. D."https://www.zbmath.org/authors/?q=ai:witherden.freddie-d"Jameson, A."https://www.zbmath.org/authors/?q=ai:jameson.antony"Vincent, P. E."https://www.zbmath.org/authors/?q=ai:vincent.peter-eSummary: This paper proposes a novel locally adaptive pseudo-time stepping convergence acceleration technique for dual time stepping which is a common integration method for solving unsteady low-Mach preconditioned/incompressible Navier-Stokes formulations. In contrast to standard local pseudo-time stepping techniques that are based on computing the local pseudo-time steps directly from estimates of the local Courant-Friedrichs-Lewy limit, the proposed technique controls the local pseudo-time steps using local truncation errors which are computed with embedded pair RK schemes. The approach has three advantages. First, it does not require an expression for the characteristic element size, which are difficult to obtain reliably for curved mixed-element meshes. Second, it allows a finer level of locality for high-order nodal discretisations, such as FR, since the local time-steps can vary between solution points and field variables. Third, it is well-suited to being combined with \(P\)-multigrid convergence acceleration. Results are presented for a laminar 2D cylinder test case at \(R e = 100\). A speed-up factor of 4.16 is achieved compared to global pseudo-time stepping with an RK4 scheme, while maintaining accuracy. When combined with \(P\)-multigrid convergence acceleration a speed-up factor of over 15 is achieved. Detailed analysis of the results reveals that pseudo-time steps adapt to element size/shape, solution state, and solution point location within each element. Finally, results are presented for a turbulent 3D SD7003 airfoil test case at \(R e = 60, 000\). Speed-ups of similar magnitude are observed, and the flow physics is found to be in good agreement with previous studies.A nonintrusive reduced order modelling approach using proper orthogonal decomposition and locally adaptive sparse grids.https://www.zbmath.org/1453.650292021-02-27T13:50:00+00:00"Alsayyari, Fahad"https://www.zbmath.org/authors/?q=ai:alsayyari.fahad"Perkó, Zoltán"https://www.zbmath.org/authors/?q=ai:perko.zoltan"Lathouwers, Danny"https://www.zbmath.org/authors/?q=ai:lathouwers.danny"Kloosterman, Jan Leen"https://www.zbmath.org/authors/?q=ai:kloosterman.jan-leenSummary: Large-scale complex systems require high fidelity models to capture the dynamics of the system accurately. The complexity of these models, however, renders their use to be expensive for applications relying on repeated evaluations, such as control, optimization, and uncertainty quantification. Proper Orthogonal Decomposition (POD) is a powerful Reduced Order Modelling (ROM) technique developed to reduce the computational burden of high fidelity models. In cases where the model is inaccessible, POD can be used in a nonintrusive manner. The accuracy and efficiency of the nonintrusive reduced model are highly dependent on the sampling scheme, especially for high dimensional problems. To that end, we study integrating the locally adaptive sparse grids with the POD method to develop a novel nonintrusive POD-based reduced order model. In our proposed approach, the locally adaptive sparse grid is used to adaptively control the sampling scheme for the POD snapshots, and the hierarchical interpolant is used as a surrogate model for the POD coefficients. An approach to efficiently update the surpluses of the sparse grids with each POD snapshots update is also introduced. The robustness and efficiency of the locally adaptive algorithm are increased by introducing a greediness parameter, and a strategy to validate the reduced model after convergence. The proposed validation algorithm can also enrich the reduced model around regions of detected discrepancies. Three numerical test cases are presented to demonstrate the potential of the proposed POD-Adaptive algorithm. The first is a nuclear reactor point kinetics, the second is a general diffusion problem, and the last is a variation of the analytical Morris function. The results show that the developed algorithm reduced the number of model evaluations compared to the classical sparse grid approach. The built reduced models captured the dynamics of the reference systems with the desired tolerances. The non-intrusiveness and simplicity of the method provide great potential for a wide range of practical large scale applications.Estimating the backward error for the least-squares problem with multiple right-hand sides.https://www.zbmath.org/1453.650792021-02-27T13:50:00+00:00"Hallman, Eric"https://www.zbmath.org/authors/?q=ai:hallman.ericSummary: Let \(A\) and \(B\) be \(m\times n\) and \(m\times d\) matrices, and let \(\widetilde{X}\) be an approximate solution to the problem \(\min_X \|AX-B\|_F\). In 1996, \textit{J.-G. Sun} [IMA J. Numer. Anal. 16, No. 1, 1--11 (1996; Zbl 0845.15002)] found an explicit expression for the \textit{optimal backward error} -- the size of the smallest perturbation to \(A\) (and possibly \(B)\) such that \(\widetilde{X}\) is an exact solution to the perturbed problem. The expression requires finding the difference of two potentially close numbers, and so its numerical evaluation can be unstable. We offer an estimate of the backward error that can be evaluated stably and when \(d=1\) is identical to the Karlson-Waldén estimate of 1997 [\textit{R. Karlson} and \textit{B. Waldén}, BIT 37, No. 4, 862--869 (1997; Zbl 0905.65051)]. We prove that this estimate always approximates the optimal backward error to within a factor of \(\sqrt{2}\).Free-stream preserving linear-upwind and WENO schemes on curvilinear grids.https://www.zbmath.org/1453.761472021-02-27T13:50:00+00:00"Zhu, Yujie"https://www.zbmath.org/authors/?q=ai:zhu.yujie"Hu, Xiangyu"https://www.zbmath.org/authors/?q=ai:hu.xiangyuSummary: Applying high-order finite-difference schemes, like the extensively used linear-upwind or WENO schemes, to curvilinear grids can be problematic. If the scheme doesn't satisfy the geometric conservation law, the geometrically induced error from grid Jacobian and metrics evaluation can pollute the flow field, and degrade the accuracy or cause the simulation failure even when uniform flow imposed, i.e. free-stream preserving problem. In order to address this issue, a method for general linear-upwind and WENO schemes preserving free-stream on stationary curvilinear grids is proposed. Following Lax-Friedrichs splitting, this method rewrites the numerical flux into a central term, which achieves free-stream preserving by using symmetrical conservative metric method, and a numerical dissipative term with a local difference form of conservative variables for neighboring grid-point pairs. In order to achieve free-stream preservation for the latter term, the local differences are modified to share the same Jacobian and metric terms evaluated by high order schemes. In addition, this method allows a simple hybridization switching between linear-upwind and WENO schemes is proposed for improving computational efficiency and reducing numerical dissipation. A number of testing cases including free-stream, isentropic vortex convection, double Mach reflection, flow past a cylinder and supersonic wind tunnel with a step are computed to verify the effectiveness of this method.Construction of periodic adapted orthonormal frames on closed space curves.https://www.zbmath.org/1453.650362021-02-27T13:50:00+00:00"Farouki, Rida T."https://www.zbmath.org/authors/?q=ai:farouki.rida-t"Kim, Soo Hyun"https://www.zbmath.org/authors/?q=ai:kim.soohyun"Moon, Hwan Pyo"https://www.zbmath.org/authors/?q=ai:pyo-moon.hwanSummary: The construction of continuous adapted orthonormal frames along \(C^1\) closed-loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid-body motions along smooth closed paths. The construction is illustrated through the simplest non-trivial context -- namely, \(C^1\) closed loops defined by a single Pythagorean-hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two-parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of \(\pi\). The desired frame is constructed through a rotation applied to the normal-plane vectors of the \textit{Euler-Rodrigues frame}, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on \(C^1\) closed-loop PH curves is possible, although this incurs transcendental terms. However, the \(C^1\) closed-loop PH quintics admit particularly simple rational periodic adapted frames.Entropy stable spectral collocation schemes for the 3-D Navier-Stokes equations on dynamic unstructured grids.https://www.zbmath.org/1453.761552021-02-27T13:50:00+00:00"Yamaleev, Nail K."https://www.zbmath.org/authors/?q=ai:yamaleev.nail-k"Del Rey Fernández, David C."https://www.zbmath.org/authors/?q=ai:del-rey-fernandez.david-c"Lou, Jialin"https://www.zbmath.org/authors/?q=ai:lou.jialin"Carpenter, Mark H."https://www.zbmath.org/authors/?q=ai:carpenter.mark-hSummary: New entropy stable spectral collocation schemes of arbitrary order of accuracy are developed for the unsteady 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. To take into account the grid motion and deformation, we use an arbitrary Lagrangian-Eulerian formulation. As a result, moving and deforming hexahedral grid elements are individually mapped onto a cube in the fixed reference system of coordinates. The proposed scheme is constructed by using the skew-symmetric form of the Navier-Stokes equations, which are discretized by using summation-by-parts spectral collocation operators that preserve the conservation properties of the original governing equations. Furthermore, the metric coefficients are approximated such that the geometric conservation laws are satisfied exactly on both static and dynamic grids. To make the scheme entropy stable, a new entropy conservative flux is derived for the 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. The new flux preserves the design order of accuracy of the original spectral collocation scheme and guarantees entropy conservation on moving and deforming grids. We present numerical results demonstrating design order of accuracy and freestream preservation properties of the new schemes for both the Euler and Navier-Stokes equations on dynamic unstructured grids.Multi-strategy based bare bones particle swarm for numerical optimisation.https://www.zbmath.org/1453.651352021-02-27T13:50:00+00:00"Liu, Jinwei"https://www.zbmath.org/authors/?q=ai:liu.jinweiSummary: Particle swarm optimisation (PSO) is a population-based stochastic search algorithm, which simulates the social behaviour of bird flocking or fish schooling. Many previous studies have shown that PSO is an effective optimisation technique in evolutionary optimisation community. However, the standard PSO still suffers from premature convergence when solving complex multimodal problems. In this paper, we propose a new PSO algorithm called multi-strategy based bare bones PSO (MPSO). The MPSO introduces generalised opposition-based learning (GOBL) and two neighbourhood search strategies into the original bare bones PSO. Simulation study is conducted on 13 well-known benchmark functions. The results show that MPSO achieves better results than the standard PSO and two other PSO algorithms.Multi-degree B-splines: algorithmic computation and properties.https://www.zbmath.org/1453.650342021-02-27T13:50:00+00:00"Toshniwal, Deepesh"https://www.zbmath.org/authors/?q=ai:toshniwal.deepesh"Speleers, Hendrik"https://www.zbmath.org/authors/?q=ai:speleers.hendrik"Hiemstra, René R."https://www.zbmath.org/authors/?q=ai:hiemstra.rene-r"Manni, Carla"https://www.zbmath.org/authors/?q=ai:manni.carla"Hughes, Thomas J. R."https://www.zbmath.org/authors/?q=ai:hughes.thomas-j-rSummary: This paper addresses theoretical considerations behind the algorithmic computation of polynomial multi-degree spline basis functions as presented in [\textit{D. Toshniwal} et al., Comput. Methods Appl. Mech. Eng. 316, 1005--1061 (2017; Zbl 1439.65016)]. The approach in [Toshniwal et al., loc. cit.] breaks from the reliance on computation of integrals recursively for building B-spline-like basis functions that span a given multi-degree spline space. The gains in efficiency are indisputable; however, the theoretical robustness needs to be examined. In this paper, we show that the construction of Toshniwal et al. [loc. cit.] yields linearly independent functions with the minimal support property that span the entire multi-degree spline space and form a convex partition of unity.Near-field imaging of inhomogeneities in a stratified ocean waveguide.https://www.zbmath.org/1453.653942021-02-27T13:50:00+00:00"Liu, Keji"https://www.zbmath.org/authors/?q=ai:liu.kejiSummary: In this work, we have derived the general representation of the scattered field in the three-layered ocean waveguide. Moreover, a generalized direct sampling method which does not rely on the equivalent condition of density, and a novel MUSIC method as an alternative approach, have been proposed for the reconstructions of penetrable obstacles in the stratified ocean waveguide. In practice, the methods are capable of recovering the scatterers of different shapes and locations, robust against noise, computationally fairly cheap and easy to carry out. We can consider them as simple and fast algorithms to supply satisfactory initial positions for the application of any existing more refined and precise but computationally more demanding techniques to achieve accurate reconstructions of physical features of objects.Numerical solution of Volterra-Fredholm integral equations using the collocation method based on a special form of the Müntz-Legendre polynomials.https://www.zbmath.org/1453.654592021-02-27T13:50:00+00:00"Negarchi, Neda"https://www.zbmath.org/authors/?q=ai:negarchi.neda"Nouri, Kazem"https://www.zbmath.org/authors/?q=ai:nouri.kazemSummary: This paper presents a computational technique based on a special family of the Müntz-Legendre polynomials to solve a class of Volterra-Fredholm integral equations. The relationship between the Jacobi polynomials and Müntz-Legendre polynomials, in a particular state, are expressed. The proposed method reduces the integral equation into algebraic equations via the Chebyshev-Gauss-Lobatto points, so that the system matrix coefficients are obtained by the least squares approximation method. The useful properties of the Jacobi polynomials are exploited to analyze the approximation error. The performance and accuracy of our method are examined with some illustrative examples.High order direct arbitrary-Lagrangian-Eulerian (ALE) \(P_NP_M\) schemes with WENO adaptive-order reconstruction on unstructured meshes.https://www.zbmath.org/1453.653092021-02-27T13:50:00+00:00"Boscheri, Walter"https://www.zbmath.org/authors/?q=ai:boscheri.walter"Balsara, Dinshaw S."https://www.zbmath.org/authors/?q=ai:balsara.dinshaw-sSummary: In this work we present a conservative WENO Adaptive Order (AO) reconstruction operator applied to an explicit one-step Arbitrary-Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) method. The spatial order of accuracy is improved by reconstructing higher order piecewise polynomials of degree \(M > N,\) starting from the underlying polynomial solution of degree \(N\) provided by the DG scheme. High order of accuracy in time is achieved by the ADER approach, making use of an element-local space-time Galerkin finite element predictor that arises from a one-step time integration procedure. As a result, space-time polynomials of order \(M + 1\) are obtained and used to perform the time evolution of the numerical solution adopting a fully explicit DG scheme. To maintain algorithm simplicity, the mesh motion is restricted to be carried out using straight lines, hence the old mesh configuration at time \(t^n\) is connected with the new one at time \(t^{n + 1}\) via space-time segments, which result in space-time control volumes on which the governing equations have to be integrated in order to obtain the time evolution of the discrete solution. Our algorithm falls into the category of \textit{direct} Arbitrary-Lagrangian-Eulerian (ALE) schemes, where the governing PDE system is directly discretized relying on a space-time conservation formulation and which already takes into account the new grid geometry \textit{directly} during the computation of the numerical fluxes. A local rezoning strategy might be used in order to locally optimize the mesh quality and avoiding the generation of invalid elements with negative determinant. The proposed approach reduces to direct ALE finite volume schemes if \(N = 0,\) while explicit direct ALE DG schemes are recovered in the case of \(N = M\). In order to stabilize the DG solution, an \textit{a priori} WENO based limiting technique is employed, that makes use of the numerical solution inside the element under consideration and its neighbor cells to find a less oscillatory polynomial approximation. By using a \textit{modal basis} in a reference element, the evaluation of the oscillation indicators is very easily and efficiently carried out, hence allowing higher order modes to be properly limited, while leaving the zero-\textit{th} order mode untouched for ensuring conservation. Numerical convergence rates for \(2 \leq N, M \leq 4\) are presented as well as a wide set of benchmark test problems for hydrodynamics on moving and fixed unstructured meshes.Structure-preserving vs. standard particle-in-cell methods: the case of an electron hybrid model.https://www.zbmath.org/1453.762322021-02-27T13:50:00+00:00"Holderied, Florian"https://www.zbmath.org/authors/?q=ai:holderied.florian"Possanner, Stefan"https://www.zbmath.org/authors/?q=ai:possanner.stefan"Ratnani, Ahmed"https://www.zbmath.org/authors/?q=ai:ratnani.ahmed"Wang, Xin"https://www.zbmath.org/authors/?q=ai:wang.xin.12|wang.xin.11|wang.xin.13|wang.xin.8|wang.xin.7|wang.xin.6|wang.xin.5|wang.xin|wang.xin.1|wang.xin.3|wang.xin.10|wang.xin.2|wang.xin.9|wang.xin.4Summary: We applied two numerical methods both belonging to the class of finite element particle-in-cell methods to a four-dimensional (one dimension in real space and three dimensions in velocity space) hybrid plasma model for electrons in a stationary, neutralizing background of ions. Here, the term \textit{hybrid} means that (energetic) electrons with velocities close to the phase velocities of the model's characteristic waves are treated kinetically, whereas electrons that are much slower than the phase velocity are treated with fluid equations. The two developed numerical schemes are based on standard finite elements on the one hand and on structure-preserving geometric finite elements on the other hand. We tested and compared the schemes in the linear and in the nonlinear stage. We show that the structure-preserving algorithm leads to better results in both stages. This can be related to the fact that the spatial discretization results in a large system of ordinary differential equations that exhibits a noncanonical Hamiltonian structure. To such systems special time integration schemes with good conservation properties can be applied.Asymptotic and positivity preserving methods for Kerr-Debye model with Lorentz dispersion in one dimension.https://www.zbmath.org/1453.780102021-02-27T13:50:00+00:00"Peng, Zhichao"https://www.zbmath.org/authors/?q=ai:peng.zhichao"Bokil, Vrushali A."https://www.zbmath.org/authors/?q=ai:bokil.vrushali-a"Cheng, Yingda"https://www.zbmath.org/authors/?q=ai:cheng.yingda"Li, Fengyan"https://www.zbmath.org/authors/?q=ai:li.fengyanSummary: In this paper, we continue our recent developments in [\textit{V. A. Bokil} et al., J. Comput. Phys. 350, 420--452 (2017; Zbl 1380.78011); J. Sci. Comput. 77, No. 1, 330--371 (2018; Zbl 1407.65094)] to devise numerical methods that have important provable properties to simulate electromagnetic wave propagation in nonlinear optical media. Particularly, we consider the one dimensional Kerr-Debye model with the Lorentz dispersion, termed as the Kerr-Debye-Lorentz model, where the nonlinearity in the polarization is a relaxed cubic Kerr type effect. The polarization also includes the linear Lorentz dispersion. As the relaxation time \(\epsilon\) goes to zero, the model will approach the Kerr-Lorentz model. The objective of this work is to devise and analyze asymptotic preserving (AP) and positivity preserving (PP) methods for the Kerr-Debye-Lorentz model. Being AP, the methods address the stiffness of the model associated with small \(\epsilon \), while capturing the correct Kerr-Lorentz limit as \(\varepsilon \to 0\) on under-resolved meshes. Being PP, the third-order nonlinear susceptibility will stay non-negative and this is important for the energy stability. In the proposed methods, the nodal discontinuous Galerkin (DG) discretizations of arbitrary order accuracy are applied in space to effectively handle nonlinearity; in time, several first and second order methods are developed. We prove that the first order in time fully discrete schemes are AP, PP and also energy stable. For the second order temporal accuracy, a novel modified exponential time integrator is proposed for the stiff part of the auxiliary differential equations modeling the electric polarization, and this is a key ingredient for the methods to be both AP and PP. In addition to a straightforward discretization of the constitutive law, we further propose a non-trivial energy-based approximation, with which the energy stability is also established mathematically. Numerical examples are presented that include an ODE example, a manufactured solution, the soliton-like propagation and the propagation of \textit{Sech} signal in fused bulk silica, to compare the proposed methods and to demonstrate the accuracy, AP and PP property. The effect of the finite relaxation time \(\epsilon\) in the model is also examined numerically.A general condition for kinetic-energy preserving discretization of flow transport equations.https://www.zbmath.org/1453.652622021-02-27T13:50:00+00:00"Veldman, Arthur E. P."https://www.zbmath.org/authors/?q=ai:veldman.arthur-eduard-paul(no abstract)A numerical method for coupling the BGK model and Euler equations through the linearized Knudsen layer.https://www.zbmath.org/1453.761492021-02-27T13:50:00+00:00"Chen, Hongxu"https://www.zbmath.org/authors/?q=ai:chen.hongxu"Li, Qin"https://www.zbmath.org/authors/?q=ai:li.qin"Lu, Jianfeng"https://www.zbmath.org/authors/?q=ai:lu.jianfengSummary: The Bhatnagar-Gross-Krook (BGK) model, a simplification of the Boltzmann equation, in the absence of boundary effect, converges to the Euler equations when the Knudsen number is small. In practice, however, Knudsen layers emerge at the physical boundary, or at the interfaces between the two regimes. We model the Knudsen layer using a half-space kinetic equation, and apply a half-space numerical solver [\textit{Q. Li} et al., ESAIM, Math. Model. Numer. Anal. 51, No. 5, 1583--1615 (2017; Zbl 1380.35006); Math. Comput. 86, No. 305, 1269--1301 (2017; Zbl 1360.35131)] to quantify the transition between the kinetic to the fluid regime. A full domain numerical solver is developed with a domain-decomposition approach, where we apply the Euler solver and kinetic solver on the appropriate subdomains and connect them via the half-space solver. In the nonlinear case, linearization is performed upon local Maxwellian. Despite the lack of analytical support, the numerical evidence nevertheless demonstrate that the linearization approach is promising.Isogeometric analysis for surface PDEs with extended loop subdivision.https://www.zbmath.org/1453.654132021-02-27T13:50:00+00:00"Pan, Qing"https://www.zbmath.org/authors/?q=ai:pan.qing"Rabczuk, Timon"https://www.zbmath.org/authors/?q=ai:rabczuk.timon"Xu, Gang"https://www.zbmath.org/authors/?q=ai:xu.gang"Chen, Chong"https://www.zbmath.org/authors/?q=ai:chen.chongSummary: We investigate the isogeometric analysis for surface PDEs based on the extended Loop subdivision approach. The basis functions consisting of quartic box-splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly, and construct the solution space for dependent variables as well, which is consistent with the concept of isogeometric analysis. The subdivision process is equivalent to the \(h\)-refinement of NURBS-based isogeometric analysis. The performance of the proposed method is evaluated by solving various surface PDEs, such as surface Laplace-Beltrami harmonic/biharmonic/triharmonic equations, which are defined on the limit surfaces of extended Loop subdivision for different initial control meshes. Numerical experiments show that the proposed method has desirable performance in terms of the accuracy, convergence and computational cost for solving the above surface PDEs defined on both open and closed surfaces. The proposed approach is proved to be second-order accuracy in the sense of \(L^2\)-norm with theoretical and/or numerical results, which is also outperformed over the standard linear finite element by several numerical comparisons.On the geometric origin of spurious waves in finite-volume discretizations of shallow water equations on triangular meshes.https://www.zbmath.org/1453.652452021-02-27T13:50:00+00:00"Danilov, S."https://www.zbmath.org/authors/?q=ai:danilov.s-d|danilov.sergey"Kutsenko, A."https://www.zbmath.org/authors/?q=ai:kutsenko.a-v|kutsenko.aleksandr|kutsenko.a-g|kutsenko.a-n|kutsenko.a-a|kutsenko.anton-aSummary: Computational wave branches are common to linearized shallow water equations discretized on triangular meshes. It is demonstrated that for standard finite-volume discretizations these branches can be traced back to the structure of the unit cell of triangular lattice, which includes two triangles with a common edge. Only subsets of similarly oriented triangles or edges possess the translational symmetry of unit cell. As a consequence, discrete degrees of freedom placed on triangles or edges are geometrically different, creating an internal structure inside unit cells. It implies a possibility of oscillations inside unit cells seen as computational branches in the framework of linearized shallow water equations, or as grid-scale noise generally. Adding dissipative operators based on smallest stencils to discretized equations is needed to control these oscillations in solutions. A review of several finite-volume discretization is presented with focus on computational branches and dissipative operators.Solving hyperbolic-elliptic problems on singular mapped disk-like domains with the method of characteristics and spline finite elements.https://www.zbmath.org/1453.653522021-02-27T13:50:00+00:00"Zoni, Edoardo"https://www.zbmath.org/authors/?q=ai:zoni.edoardo"Güçlü, Yaman"https://www.zbmath.org/authors/?q=ai:guclu.yamanSummary: A common strategy in the numerical solution of partial differential equations is to define a uniform discretization of a tensor-product multi-dimensional logical domain, which is mapped to a physical domain through a given coordinate transformation. By extending this concept to a multi-patch setting, simple and efficient numerical algorithms can be employed on relatively complex geometries. The main drawback of such an approach is the inherent difficulty in dealing with singularities of the coordinate transformation. This work suggests a comprehensive numerical strategy for the common situation of disk-like domains with a singularity at a unique pole, where one edge of the rectangular logical domain collapses to one point of the physical domain (for example, a circle). We present robust numerical methods for the solution of Vlasov-like hyperbolic equations coupled to Poisson-like elliptic equations in such geometries. We describe a semi-Lagrangian advection solver that employs a novel set of coordinates, named pseudo-Cartesian coordinates, to integrate the characteristic equations in the whole domain, including the pole, and a finite element elliptic solver based on globally \(\mathcal{C}^1\) smooth splines [\textit{D. Toshniwal} et al., Comput. Methods Appl. Mech. Eng. 316, 1005--1061 (2017; Zbl 1439.65016)]. The two solvers are tested both independently and on a coupled model, namely the 2D guiding-center model for magnetized plasmas, equivalent to a vorticity model for incompressible inviscid Euler fluids. The numerical methods presented show high-order convergence in the space discretization parameters, uniformly across the computational domain, without effects of order reduction due to the singularity. Dedicated tests show that the numerical techniques described can be applied straightforwardly also in the presence of point charges (equivalently, point-like vortices), within the context of particle-in-cell methods.Collocation of next-generation operators for computing the basic reproduction number of structured populations.https://www.zbmath.org/1453.920062021-02-27T13:50:00+00:00"Breda, Dimitri"https://www.zbmath.org/authors/?q=ai:breda.dimitri"Kuniya, Toshikazu"https://www.zbmath.org/authors/?q=ai:kuniya.toshikazu"Ripoll, Jordi"https://www.zbmath.org/authors/?q=ai:ripoll.jordi"Vermiglio, Rossana"https://www.zbmath.org/authors/?q=ai:vermiglio.rossanaSummary: We contribute a full analysis of theoretical and numerical aspects of the collocation approach recently proposed by some of the authors to compute the basic reproduction number of structured population dynamics as spectral radius of certain infinite-dimensional operators. On the one hand, we prove under mild regularity assumptions on the models coefficients that the concerned operators are compact, so that the problem can be properly recast as an eigenvalue problem thus allowing for numerical discretization. On the other hand, we prove through detailed and rigorous error and convergence analyses that the method performs the expected spectral accuracy. Several numerical tests validate the proposed analysis by highlighting diverse peculiarities of the investigated approach.Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations.https://www.zbmath.org/1453.653502021-02-27T13:50:00+00:00"Zhou, Boya"https://www.zbmath.org/authors/?q=ai:zhou.boya"Chen, Xiaoli"https://www.zbmath.org/authors/?q=ai:chen.xiaoli"Li, Dongfang"https://www.zbmath.org/authors/?q=ai:li.dongfangSummary: The solutions of the nonlinear time fractional parabolic problems usually undergo dramatic changes at the beginning. In order to overcome the initial singularity, the temporal discretization is done by using the Alikhanov schemes on the nonuniform meshes. And the spatial discretization is achieved by using the finite element methods. The optimal error estimates of the fully discrete schemes hold without certain time-step restrictions dependent on the spatial mesh sizes. Such unconditionally optimal convergent results are proved by taking the global behavior of the analytical solutions into account. Numerical results are presented to confirm the theoretical findings.Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime.https://www.zbmath.org/1453.652782021-02-27T13:50:00+00:00"Bao, Weizhu"https://www.zbmath.org/authors/?q=ai:bao.weizhu"Zhao, Xiaofei"https://www.zbmath.org/authors/?q=ai:zhao.xiaofeiSummary: Different efficient and accurate numerical methods have recently been proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter \(\varepsilon \in(0, 1],\) which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. \(0 < \varepsilon \ll 1\), the solution of the NKGE propagates waves with wavelength at \(O(1)\) and \(O(\varepsilon^2)\) in space and time, respectively, which brings significantly numerical burdens in designing numerical methods. We compare systematically spatial/temporal efficiency and accuracy as well as \(\epsilon\)-resolution (or \(\epsilon\)-scalability) of different numerical methods including finite difference time domain methods, time-splitting method, exponential wave integrator, limit integrator, multiscale time integrator, two-scale formulation method and iterative exponential integrator. Finally, we adopt the multiscale time integrator to study the convergence rates from the NKGE to its limiting models when \(\varepsilon \to 0^+\).A distributed active subspace method for scalable surrogate modeling of function valued outputs.https://www.zbmath.org/1453.650092021-02-27T13:50:00+00:00"Guy, Hayley"https://www.zbmath.org/authors/?q=ai:guy.hayley"Alexanderian, Alen"https://www.zbmath.org/authors/?q=ai:alexanderian.alen"Yu, Meilin"https://www.zbmath.org/authors/?q=ai:yu.meilinSummary: We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen-Loève (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport.Numerical inverse scattering for the sine-Gordon equation.https://www.zbmath.org/1453.652902021-02-27T13:50:00+00:00"Deconinck, Bernard"https://www.zbmath.org/authors/?q=ai:deconinck.bernard"Trogdon, Thomas"https://www.zbmath.org/authors/?q=ai:trogdon.thomas"Yang, Xin"https://www.zbmath.org/authors/?q=ai:yang.xinSummary: We implement the numerical inverse scattering transform (NIST) for the sine-Gordon equation in laboratory coordinates on the real line using the method developed by [\textit{T. Trogdon} and \textit{S. Olver}, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 469, No. 2149, Article ID 20120330, 22 p. (2013; Zbl 1372.65356)]. The NIST allows one to compute the solution at any \(x\) and \(t\) without having spatial discretization or time-stepping. The numerical implementation is fully spectrally accurate. With the help of the method of nonlinear steepest descent, the NIST is demonstrated to be uniformly accurate.A parallelized computational model for multidimensional systems of coupled nonlinear fractional hyperbolic equations.https://www.zbmath.org/1453.652272021-02-27T13:50:00+00:00"Macías-Díaz, J. E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardoSummary: In this work, we consider a general multidimensional system of hyperbolic partial differential equations with fractional diffusion of the Riesz type, constant damping and coupled nonlinear reaction terms. The system generalizes many particular models from the physical sciences (including inhibitor-activator models in chemistry, diffusive nonlinear systems in population dynamics and relativistic wave equations), and considers the presence of an arbitrary number of both spatial dimensions and dependent variables. Motivated by the wide range of applications, we propose an explicit four-step finite-difference methodology to approximate the solutions of the continuous system. The properties of stability, boundedness and convergence of the scheme are proved rigorously using a discrete form of the fractional energy method. An efficient computational implementation of the scheme is also proposed in this work. It is important to recall that algorithms for space-fractional systems are computationally highly demanding. To alleviate this problem, a parallel implementation of our scheme is proposed using a vector reformulation of the numerical method. We provide some illustrative simulations on the formation of complex patterns in the two-dimensional scenario, and even in the computationally intense three-dimensional case. For the sake of convenience, an algorithmic presentation of our computational model is provided in this manuscript.Variable smoothing for convex optimization problems using stochastic gradients.https://www.zbmath.org/1453.901182021-02-27T13:50:00+00:00"Boţ, Radu Ioan"https://www.zbmath.org/authors/?q=ai:bot.radu-ioan"Böhm, Axel"https://www.zbmath.org/authors/?q=ai:bohm.axelSummary: We aim to solve a structured convex optimization problem, where a nonsmooth function is composed with a linear operator. When opting for full splitting schemes, usually, primal-dual type methods are employed as they are effective and also well studied. However, under the additional assumption of Lipschitz continuity of the nonsmooth function which is composed with the linear operator we can derive novel algorithms through regularization via the Moreau envelope. Furthermore, we tackle large scale problems by means of stochastic oracle calls, very similar to stochastic gradient techniques. Applications to total variational denoising and deblurring, and matrix factorization are provided.Zero-flux approximations for multivariate quadrature-based moment methods.https://www.zbmath.org/1453.653742021-02-27T13:50:00+00:00"Pollack, Martin"https://www.zbmath.org/authors/?q=ai:pollack.martin"Pütz, Michele"https://www.zbmath.org/authors/?q=ai:putz.michele"Marchisio, Daniele L."https://www.zbmath.org/authors/?q=ai:marchisio.daniele-l"Oevermann, Michael"https://www.zbmath.org/authors/?q=ai:oevermann.michael"Hasse, Christian"https://www.zbmath.org/authors/?q=ai:hasse.christianSummary: The evolution of polydisperse systems is governed by population balance equations. A group of efficient solution approaches are the moment methods, which do not solve for the number density function (NDF) directly but rather for a set of its moments. While this is computationally efficient, a specific challenge arises when describing the fluxes across a boundary in phase space for the disappearance of elements, the so-called zero-flux. The main difficulty is the missing NDF-information at the boundary, which most moment methods cannot provide. Relevant physical examples are evaporating droplets, soot oxidation or particle dissolution. In general, this issue can be solved by reconstructing the NDF close to the boundary. However, this was previously only achieved with univariate approaches, i.e. considering only a single internal variable. Many physical problems are insufficiently described by univariate population balance equations, e.g. droplets in sprays often require the temperature or the velocity to be internal coordinates in addition to the size. In this paper, we propose an algorithm, which provides an efficient multivariate approach to calculate the zero-fluxes. The algorithm employs the Extended Quadrature Method of Moments (EQMOM) with Beta and Gamma kernel density functions for the marginal NDF reconstruction and a polynomial or spline for the other conditional dimensions. This combination allows to reconstruct the entire multivariate NDF and based on this, expressions for the disappearance flux are derived. An algorithm is proposed for the whole moment inversion and reconstruction process. It is validated against a suite of test cases with increasing complexity. The influence of the number of kernel density functions and the configuration of the polynomials and splines on the accuracy is discussed. Finally, the associated computational costs are evaluated.A perfectly matched layer formulation adapted for fast frequency sweeps of exterior acoustics finite element models.https://www.zbmath.org/1453.654182021-02-27T13:50:00+00:00"Vermeil de Conchard, Antoine"https://www.zbmath.org/authors/?q=ai:vermeil-de-conchard.antoine"Mao, Huina"https://www.zbmath.org/authors/?q=ai:mao.huina"Rumpler, Romain"https://www.zbmath.org/authors/?q=ai:rumpler.romainSummary: Effective treatment of unbounded domains using artificial truncating boundaries are essential in numerical simulation, e.g. using the Finite Element Method (FEM). Among these, Perfectly Matched Layers (PML) have proved to be particularly efficient and flexible. However, an efficient handling of frequency sweeps is not trivial with such absorbing layers since the formulation inherently contains coupled space- and frequency-dependent terms. Using the FEM, this may imply generating system matrices at each step of the frequency sweep. In this paper, an approximation is proposed in order to allow for efficient frequency sweeps. The performance and robustness of the proposed approximation is presented on 2D and 3D acoustic cases. A generic, robust way to truncate the acoustic domain efficiently is also proposed, tested on a range of test cases and for different frequency regions.A double extrapolation primal-dual algorithm for saddle point problems.https://www.zbmath.org/1453.651462021-02-27T13:50:00+00:00"Wang, Kai"https://www.zbmath.org/authors/?q=ai:wang.kai.1"He, Hongjin"https://www.zbmath.org/authors/?q=ai:he.hongjinSummary: The first-order primal-dual algorithms have received much considerable attention in the literature due to their quite promising performance in solving large-scale image processing models. In this paper, we consider a general saddle point problem and propose a double extrapolation primal-dual algorithm, which employs the efficient extrapolation strategy for both primal and dual variables. It is remarkable that the proposed algorithm enjoys a unified framework including several existing efficient solvers as special cases. Another exciting property is that, under quite flexible requirements on the involved extrapolation parameters, our algorithm is globally convergent to a saddle point of the problem under consideration. Moreover, the worst case \(\mathcal{O}(1/t)\) convergence rate in both ergodic and nonergodic senses, and the linear convergence rate can be established for more general cases, where \(t\) counts the iteration. Some computational results on solving image deblurring, image inpainting and the nearest correlation matrix problems further show that the proposed algorithm is efficient, and performs better than some existing first-order solvers in terms of taking less iterations and computing time in some cases.Using Oshima splines to produce accurate numerical results and high quality graphical output.https://www.zbmath.org/1453.682252021-02-27T13:50:00+00:00"Takato, Setsuo"https://www.zbmath.org/authors/?q=ai:takato.setsuo"Vallejo, José A."https://www.zbmath.org/authors/?q=ai:vallejo.jose-antonioSummary: We illustrate the use of Oshima splines in producing high-quality \LaTeX output in two cases: first, the numerical computation of derivatives and integrals, and second, the display of silhouettes and wireframe surfaces, using the macros package KeTCindy. Both cases are of particular interest for college and university teachers wanting to create handouts to be used by students, or drawing figures for a research paper. When dealing with numerical computations, KeTCindy can make a call to the CAS Maxima to check for accuracy; in the case of surface graphics, it is particularly important to be able to detect intersections of projected curves, and we show how to do it in a seamlessly manner using Oshima splines in KeTCindy. A C compiler can be called in this case to speed up computations.The compact gradient recovery discontinuous Galerkin method for diffusion problems.https://www.zbmath.org/1453.760732021-02-27T13:50:00+00:00"Johnson, Philip E."https://www.zbmath.org/authors/?q=ai:johnson.philip-e"Johnsen, Eric"https://www.zbmath.org/authors/?q=ai:johnsen.ericSummary: We introduce a new family of schemes, labeled Compact Gradient Recovery (CGR), for handling parabolic partial differential equations within the discontinuous Galerkin (DG) framework, with as the ultimate goal the discretization of diffusive flux terms in advection-diffusion systems such as the compressible Navier-Stokes equations. Like other DG approaches for diffusion, this family of schemes is based on the mixed formulation, where an auxiliary variable is introduced (but not stored between timesteps) to approximate the solution gradient. To maximize accuracy, wherever interface approximations are necessary within the mixed formulation, our schemes apply the Recovery concept originally introduced by \textit{B. Van Leer} and \textit{S. Nomura} [``Discontinuous Galerkin for diffusion'', AIAA Paper 2005-5108, \url{doi:10.2514/6.2005-5108}]. However, unlike Recovery DG, our new family of schemes is based on a nearest-neighbor stencil and does not require differentiation of the recovered solution, which can introduce large errors for shear diffusion problems. By design, the new schemes fill the void between the highly accurate Recovery DG schemes and existing mixed-form and penalty-based formulations. Fourier analysis is performed on Cartesian and simplex meshes to determine the order of accuracy and timestep-size stability limits for different solution polynomial orders \(p\). Similar to other DG approaches for diffusion, our proposed schemes make use of a jump penalization factor when solving for the gradient along interfaces; the effect of this parameter is explored through Fourier analysis, allowing an informed recommendation regarding its value. In addition to exploration of the new scheme, the analysis includes previously unknown properties of the widely used BR2 scheme under a broad set of configurations. Our new approach is verified using a comprehensive suite of test problems (scalar Laplacian diffusion, shear diffusion, and compressible Navier-Stokes) on different mesh types and compared to the gold-standard, nearest-neighbor approach for diffusion, the BR2 scheme; our new scheme performs favorably with regard to accuracy of the solution, accuracy of the solution gradient, and timestep size without a substantial increase in computational cost.Scaling to the stars -- a linearly scaling elliptic solver for \(p\)-multigrid.https://www.zbmath.org/1453.761522021-02-27T13:50:00+00:00"Huismann, Immo"https://www.zbmath.org/authors/?q=ai:huismann.immo"Stiller, Jörg"https://www.zbmath.org/authors/?q=ai:stiller.jorg"Fröhlich, Jochen"https://www.zbmath.org/authors/?q=ai:frohlich.jochenSummary: High-order methods gain increased attention in computational fluid dynamics. However, due to the time step restrictions arising from the semi-implicit time stepping for the incompressible case, the potential advantage of these methods depends critically on efficient elliptic solvers. Due to the operation counts of operators scaling with the polynomial degree \(p\) times the number of degrees of freedom \(n_{\operatorname{DOF}},\) the runtime of the best available multigrid solvers scales with \(\mathcal{O}(p \cdot n_{\operatorname{DOF}})\). This scaling with \(p\) significantly lowers the applicability of high-order methods to high orders. While the operators for residual evaluation can be linearized when using static condensation, \textsc{Schwarz}-type smoothers require their inverses on fixed subdomains. No explicit inverse is known in the condensed case and matrix-matrix multiplications scale with \(p \cdot n_{\operatorname{DOF}}\). This paper derives a matrix-free explicit inverse for the static condensed operator in a cuboidal, Cartesian subdomain. It scales with \(p^3\) per element, i.e.\( n_{\operatorname{DOF}}\) globally, and allows for a linearly scaling additive \textsc{Schwarz} smoother, yielding a \(p\)-multigrid cycle with an operation count of \(\mathcal{O}(n_{\operatorname{DOF}})\). The resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with \(n_{\operatorname{DOF}}\) for polynomial degrees at least up to 48. Furthermore the runtime is less than one microsecond per unknown over wide parameter ranges when using one core of a CPU, leading to time-stepping for the incompressible \textsc{Navier-Stokes} equations using as much time for explicitly treated convection terms as for the elliptic solvers.Iterative solution and preconditioning for the tangent plane scheme in computational micromagnetics.https://www.zbmath.org/1453.821022021-02-27T13:50:00+00:00"Kraus, Johannes"https://www.zbmath.org/authors/?q=ai:kraus.johannes-k"Pfeiler, Carl-Martin"https://www.zbmath.org/authors/?q=ai:pfeiler.carl-martin"Praetorius, Dirk"https://www.zbmath.org/authors/?q=ai:praetorius.dirk"Ruggeri, Michele"https://www.zbmath.org/authors/?q=ai:ruggeri.michele"Stiftner, Bernhard"https://www.zbmath.org/authors/?q=ai:stiftner.bernhardSummary: The tangent plane scheme is a time-marching scheme for the numerical solution of the nonlinear parabolic Landau-Lifshitz-Gilbert equation, which describes the time evolution of ferromagnetic configurations. Exploiting the geometric structure of the equation, the tangent plane scheme requires only the solution of one linear variational form per time-step, which is posed in the discrete tangent space determined by the nodal values of the current magnetization. We develop an effective solution strategy for the arising constrained linear systems, which is based on appropriate Householder reflections. We derive possible preconditioners, which are (essentially) independent of the time-step, and prove linear convergence of the preconditioned GMRES algorithm. Numerical experiments underpin the theoretical findings.Diagnosing numerical Cherenkov instabilities in relativistic plasma simulations based on general meshes.https://www.zbmath.org/1453.653412021-02-27T13:50:00+00:00"Na, D.-Y."https://www.zbmath.org/authors/?q=ai:na.dong-yeop"Nicolini, J. L."https://www.zbmath.org/authors/?q=ai:nicolini.j-l"Lee, R."https://www.zbmath.org/authors/?q=ai:lee.ronnie|lee.ritchie|lee.royal|lee.rosanna|lee.ryan|lee.regina|lee.russell|lee.ryeong"Borges, B.-H. V."https://www.zbmath.org/authors/?q=ai:borges.ben-hur-v"Omelchenko, Y. A."https://www.zbmath.org/authors/?q=ai:omelchenko.yuri-a"Teixeira, F. L."https://www.zbmath.org/authors/?q=ai:teixeira.fernando-lSummary: Numerical Cherenkov radiation (NCR) or instability is a detrimental effect frequently found in electromagnetic particle-in-cell (EM-PIC) simulations involving relativistic plasma beams. NCR is caused by spurious coupling between electromagnetic-field modes and multiple beam resonances. This coupling may result from the slow down of poorly-resolved waves due to numerical (grid) dispersion and from aliasing mechanisms. NCR has been studied in the past for finite-difference-based EM-PIC algorithms on regular (structured) meshes with rectangular elements. In this work, we extend the analysis of NCR to finite-element-based EM-PIC algorithms implemented on unstructured meshes. The influence of different mesh element shapes and mesh layouts on NCR is studied. Analytic predictions are compared against results from finite-element-based EM-PIC simulations of relativistic plasma beams on various mesh types.Smoothing and parameter estimation by soft-adherence to governing equations.https://www.zbmath.org/1453.623992021-02-27T13:50:00+00:00"Rudy, Samuel H."https://www.zbmath.org/authors/?q=ai:rudy.samuel-h"Brunton, Steven L."https://www.zbmath.org/authors/?q=ai:brunton.steven-l"Kutz, J. Nathan"https://www.zbmath.org/authors/?q=ai:kutz.j-nathanSummary: The analysis of high-dimensional dynamical systems generally requires the integration of simulation data with experimental measurements. Experimental data often has substantial amounts of measurement noise that compromises the ability to produce accurate dimensionality reduction, parameter estimation, reduced order models, and/or balanced models for control. Data assimilation attempts to overcome the deleterious effects of noise by producing a set of algorithms for state estimation from noisy and possibly incomplete measurements. Indeed, methods such as Kalman filtering and smoothing are vital tools for scientists in fields ranging from electronics to weather forecasting. In this work we develop a novel framework for smoothing data based on known or partially known nonlinear governing equations. The method yields superior results to current techniques when applied to problems with known deterministic dynamics. By exploiting the numerical time-stepping constraints of the deterministic system, an optimization formulation can readily extract the noise from the nonlinear dynamics in a principled manner. The superior performance is due in part to the fact that it optimizes global state estimates. We demonstrate the efficiency and efficacy of the method on a number of canonical examples, thus demonstrating its viability for the wide range of potential applications stated above.Advection without compounding errors through flow map composition.https://www.zbmath.org/1453.652542021-02-27T13:50:00+00:00"Kulkarni, Chinmay S."https://www.zbmath.org/authors/?q=ai:kulkarni.chinmay-s"Lermusiaux, Pierre F. J."https://www.zbmath.org/authors/?q=ai:lermusiaux.pierre-f-jSummary: We propose a novel numerical methodology to compute the advective transport and diffusion-reaction of tracer quantities. The tracer advection occurs through flow map composition and is super-accurate, yielding numerical solutions almost devoid of compounding numerical errors, while allowing for direct parallelization in the temporal direction. It is computed by implicitly solving the characteristic evolution through a modified transport partial differential equation and domain decomposition in the temporal direction, followed by composition with the known initial condition. This advection scheme allows a rigorous computation of the spatial and temporal error bounds, yields an accuracy comparable to that of Lagrangian methods, and maintains the advantages of Eulerian schemes. We further show that there exists an optimal value of the composition timestep that yields the minimum total numerical error in the computations, and derive the expression for this value. We develop schemes for the addition of tracer diffusion, reaction, and source terms, and for the implementation of boundary conditions. Finally, the methodology is applied in three flow examples, namely an analytical reversible swirl flow, an idealized flow exiting a strait undergoing sudden expansion, and a realistic ocean flow in the Bismarck sea. New benchmark problems for advection-diffusion-reaction schemes are developed and used to compare and contrast results with those of classic schemes. The results highlight the theoretical properties of the methodology as well as its efficiency, super-accuracy with minimal numerical errors, and applicability in realistic simulations.Finite difference methods for Caputo-Hadamard fractional differential equations.https://www.zbmath.org/1453.651862021-02-27T13:50:00+00:00"Gohar, Madiha"https://www.zbmath.org/authors/?q=ai:gohar.madiha"Li, Changpin"https://www.zbmath.org/authors/?q=ai:li.changpin|li.changpin.1"Li, Zhiqiang"https://www.zbmath.org/authors/?q=ai:li.zhiqiang|li.zhiqiang.1Summary: In this paper, we study finite difference methods for fractional differential equations (FDEs) with Caputo-Hadamard derivatives. First, smoothness properties of the solution are investigated. The fractional rectangular, \(L_{\log,1}\) interpolation, and modified predictor-corrector methods for Caputo-Hadamard fractional ordinary differential equations (FODEs) are proposed through approximating the corresponding equivalent Volterra integral equations. The stability and error estimate of the derived methods are proved as well. Then, we investigate finite difference methods for fractional partial differential equations (FPDEs) with Caputo-Hadamard derivative. By applying the constructed \(L1\) scheme for approximating the time fractional derivative, a semi-discrete difference scheme is derived. The stability and convergence analysis are shown too in detail. Furthermore, a fully discrete scheme is established by the standard second-order difference scheme in spacial direction. Stability and error estimate are also presented. The numerical experiments are displayed to verify the theoretical results.A fast preconditioned iterative method for the electromagnetic scattering by multiple cavities with high wave numbers.https://www.zbmath.org/1453.780152021-02-27T13:50:00+00:00"Zhao, Meiling"https://www.zbmath.org/authors/?q=ai:zhao.meiling"Zhu, Na"https://www.zbmath.org/authors/?q=ai:zhu.naSummary: We study the problem of electromagnetic scattering by multiple open cavities embedded in an infinite ground plane with high wave numbers. The problem can be described by a series of Helmholtz equations with coupled boundary conditions. We develop a sixth-order finite difference scheme to discretize the coupled Helmholtz equations. By Gaussian elimination in the vertical direction and Fourier transform in the horizontal direction, we can reduce the multiple cavity scattering problem to an aperture linear system. However, in the situation of high wave numbers, the condition number of the coefficient matrix of the reduced linear system is especially large and the system tends to be ill-conditioned. The convergence histories of most iterative methods become oscillating which consume considerable computations and memory spaces. In order to overcome the difficulty caused by high wave numbers, we develop an efficient preconditioned iterative method based on the Krylov subspace, which greatly improves the eigenvalue distributions and reduces the number of iterations. Numerical experiments show the validity and efficiency of the proposed sixth-order fast preconditioned algorithm for solving the scattering by multiple cavities with high wave numbers.Fully-conservative contact-capturing schemes for multi-material advection.https://www.zbmath.org/1453.761152021-02-27T13:50:00+00:00"Williams, R. J. R."https://www.zbmath.org/authors/?q=ai:williams.robin-j-rSummary: We present a number of fully-conservative, four-equation, multicomponent reconstruction schemes for the Euler equations, which have been designed to maintain both constant pressure and temperature at isothermal contact discontinuities, as is required by thermodynamic consistency. The schemes we discuss have been implemented in AWE's staggered-mesh detailed turbulence modelling code \textsc{TURMOIL} and also in a finite-volume Riemann-solver based code. We show results for a problem where material properties with a high contrast in adiabatic index and material specific heat lead to substantial pressure and temperature perturbations for a simple mass fraction model. The improved schemes maintain the isobaric and isothermal nature of the initial condition to numerical precision.Near critical, self-similar, blow-up solutions of the generalised Korteweg-de Vries equation: asymptotics and computations.https://www.zbmath.org/1453.370642021-02-27T13:50:00+00:00"Amodio, Pierluigi"https://www.zbmath.org/authors/?q=ai:amodio.pierluigi"Budd, Chris J."https://www.zbmath.org/authors/?q=ai:budd.chris-j"Koch, Othmar"https://www.zbmath.org/authors/?q=ai:koch.othmar"Rottschäfer, Vivi"https://www.zbmath.org/authors/?q=ai:rottschafer.vivi"Settanni, Giuseppina"https://www.zbmath.org/authors/?q=ai:settanni.giuseppina"Weinmüller, Ewa"https://www.zbmath.org/authors/?q=ai:weinmuller.ewa-bSummary: In this article we give a detailed asymptotic analysis of the near critical self-similar blowup solutions to the Generalised Korteweg-de Vries equation (GKdV). We compare this analysis to some careful numerical calculations. It has been known that for a nonlinearity that has a power larger than the critical value \(p = 5\), solitary waves of the GKdV can become unstable and become infinite in finite time, in other words they blow up. Numerical simulations presented in [\textit{C. Klein} and \textit{R. Peter}, Physica D 304--305, 52--78 (2015; Zbl 1364.65182)] indicate that if \(p > 5\) the solitary waves travel to the right with an increasing speed, and simultaneously, form a similarity structure as they approach the blow-up time. This structure breaks down at \(p = 5\). Based on these observations, we rescale the GKdV equation to give an equation that will be analysed by using asymptotic methods as \(p \rightarrow 5^+\). By doing this we resolve the complete structure of these self-similar blow-up solutions and study the singular nature of the solutions in the critical limit. In both the numerics and the asymptotics, we find that the solution has sech-like behaviour near the peak. Moreover, it becomes asymmetric with slow algebraic decay to the left of the peak and much more rapid algebraic decay to the right. The asymptotic expressions agree to high accuracy with the numerical results, performed by adaptive high-order solvers based on collocation or finite difference methods.Fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices.https://www.zbmath.org/1453.654432021-02-27T13:50:00+00:00"Li, Yu"https://www.zbmath.org/authors/?q=ai:li.yu"Slevinsky, Richard Mikaël"https://www.zbmath.org/authors/?q=ai:slevinsky.richard-mikaelSummary: We present a unified treatment of the Fourier spectra of spherically symmetric nonlocal diffusion operators. We develop numerical and analytical results for the class of kernels with weak algebraic singularity as the distance between source and target tends to 0. Rapid algorithms are derived for their Fourier spectra with the computation of each eigenvalue independent of all others. The algorithms are trivially parallelizable, capable of leveraging more powerful compute environments, and the accuracy of the eigenvalues is individually controllable. The algorithms include a Maclaurin series and a full divergent asymptotic series valid for any \(d\) spatial dimensions. Using Drummond's sequence transformation, we prove linear complexity recurrence relations for degree-graded sequences of numerators and denominators in the rational approximations to the divergent asymptotic series. These relations are important to ensure that the algorithms are efficient, and also increase the numerical stability compared with the conventional algorithm with quadratic complexity.Imposing jump conditions on nonconforming interfaces for the correction function method: a least squares approach.https://www.zbmath.org/1453.653842021-02-27T13:50:00+00:00"Marques, Alexandre Noll"https://www.zbmath.org/authors/?q=ai:marques.alexandre-noll"Nave, Jean-Christophe"https://www.zbmath.org/authors/?q=ai:nave.jean-christophe"Rosales, Rodolfo Ruben"https://www.zbmath.org/authors/?q=ai:rosales.rodolfo-rubenSummary: We introduce a technique that simplifies the problem of imposing jump conditions on interfaces that are not aligned with a computational grid in the context of the \textit{Correction Function Method} (CFM). The CFM offers a general framework to solve Poisson's equation in the presence of discontinuities to high order of accuracy, while using a compact discretization stencil. A key concept behind the CFM is enforcing the jump conditions in a least squares sense. This concept requires computing integrals over sections of the interface, which is a challenge in 3-D when only an implicit representation of the interface is available (e.g., the zero contour of a level set function). The technique introduced here is based on a new formulation of the least squares procedure that relies only on integrals over domains that are amenable to simple quadrature after local coordinate transformations. We incorporate this technique into a fourth order accurate implementation of the CFM, and show examples of solutions to Poisson's equation with imposed jump conditions computed in 2-D and 3-D.Numerical scheme for kinetic transport equation with internal state.https://www.zbmath.org/1453.652612021-02-27T13:50:00+00:00"Vauchelet, Nicolas"https://www.zbmath.org/authors/?q=ai:vauchelet.nicolas"Yasuda, Shugo"https://www.zbmath.org/authors/?q=ai:yasuda.shugoA discontinuous Galerkin method for wave propagation in orthotropic poroelastic media with memory terms.https://www.zbmath.org/1453.740322021-02-27T13:50:00+00:00"Xie, Jiangming"https://www.zbmath.org/authors/?q=ai:xie.jiangming"Ou, M. Yvonne"https://www.zbmath.org/authors/?q=ai:ou.miao-jung-yvonne"Xu, Liwei"https://www.zbmath.org/authors/?q=ai:xu.liweiSummary: Poroelastic materials play an important role in biomechanical and geophysical research. In this paper, we investigate wave propagation in orthotropic poroelastic media by studying the time-domain poroelastic wave equations. Both the low frequency Biot's (LF-Biot) equations and the Biot-Johnson-Koplik-Dashen (Biot-JKD) model are considered. In LF-Biot equations, the dissipation terms are proportional to the relative velocity between the fluid and the solid by a constant. Contrast to this, the dissipation terms in the Biot-JKD model are in the form of time convolution (memory) as a result of the frequency-dependence of fluid-solid interaction at the underlying microscopic scale in the frequency domain. The dynamic tortuosity and permeability described by Darcy's law are two crucial factors in this problem, and highly linked to the viscous force. In the Biot model, the key difficulty is to handle the viscous term when the pore fluid is viscous. In the Biot-JKD model, the convolution operator involves order 1/2 shifted fractional derivatives in the time domain, which is challenging to discretize. In this work, a new method of the multipoint Padé (or Rational) approximation for Stieltjes function is applied to approximate the JKD dynamic tortuosity and then an augmented system of Biot-JKD model is obtained, where the kernel of the memory term is replaced by the finite auxiliary variables satisfying a local system of ordinary differential equations. The Runge-Kutta discontinuous Galerkin (RKDG) method with the un-splitting method is used to compute the numerical solution, and numerical examples are presented to demonstrate the high order accuracy and stability of the method. Compared with the existing approaches for solving the Biot-JKD equations, the augmented system presented here require neither the storage of solution history nor the computation of the flux of the auxiliary variables.Positivity preserving finite difference methods for Poisson-Nernst-Planck equations with steric interactions: application to slit-shaped nanopore conductance.https://www.zbmath.org/1453.652132021-02-27T13:50:00+00:00"Ding, Jie"https://www.zbmath.org/authors/?q=ai:ding.jie"Wang, Zhongming"https://www.zbmath.org/authors/?q=ai:wang.zhongming"Zhou, Shenggao"https://www.zbmath.org/authors/?q=ai:zhou.shenggaoSummary: To study ion transport in electrolyte solutions, we propose numerical methods for a modified Poisson-Nernst-Planck model with ionic steric effects (SPNP). Positivity preserving schemes based on harmonic-mean approximations are proposed on a nonuniform mesh for the spatial discretization of the SPNP equations. Both explicit and semi-implicit discretization are considered in time. Numerical analysis shows that explicit forward Euler and semi-implicit trapezoidal discretization lead to schemes that maintain fully discrete solution positivity under a constraint on a mesh ratio, while the semi-implicit backward Euler discretization maintain fully discrete solution positivity \textit{unconditionally}. We further study the condition numbers of the matrix associated with the semi-implicit backward Euler discretization, and establish an upper bound on condition numbers, indicating that the developed discretization based on harmonic-mean approximations effectively solves the issue of the presence of large condition numbers when using the Slotboom-type variables. Further numerical simulations confirm the analysis results on accuracy, positivity, and bounded condition numbers. The proposed schemes are also applied to study practical applications, such as the impact of self and cross steric interactions on ion distribution and rectifying behavior in a slit-shaped nanopore with surface charges. Possible extensions of the numerical methods to other modified PNP models with ion correlations and steric effects are also discussed.Algorithm for some anomalously diffusive hyperbolic systems in molecular dynamics: theoretical analysis and pattern formation.https://www.zbmath.org/1453.652742021-02-27T13:50:00+00:00"Macías-Díaz, J. E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardo"Hendy, A. S."https://www.zbmath.org/authors/?q=ai:hendy.ahmed-sSummary: Departing from a two-dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions, we investigate a general form of that model using a finite-difference approach. The model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. The presence of two-dimensional diffusive terms consisting of fractional operators of the Riesz type is considered here, using spatial differentiation orders in the set \((0, 1) \cup(1, 2]\). We impose initial conditions on a closed and bounded rectangle, and a finite-difference methodology based on the use of fractional centered differences is proposed. Among the most important results of this work, we prove the existence and the uniqueness of the solutions of the numerical method, and establish analytically the second-order consistency of our scheme. Moreover, the discrete energy method is employed to prove the stability and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the presence of Turing patterns and wave instabilities, in agreement with some reports found in the literature on superdiffusive hyperbolic activator-inhibitor systems. We show numerically that the presence of Turing patterns is independent of the size of the spatial domain. As a new application, we show that Turing patterns are also present in subdiffusive scenarios.Hadamard-Babich ansatz for point-source elastic wave equations in variable media at high frequencies.https://www.zbmath.org/1453.654152021-02-27T13:50:00+00:00"Qian, Jianliang"https://www.zbmath.org/authors/?q=ai:qian.jianliang"Song, Jian"https://www.zbmath.org/authors/?q=ai:song.jian.1|song.jian.2|song.jian|song.jian.3"Lu, Wangtao"https://www.zbmath.org/authors/?q=ai:lu.wangtao"Burridge, Robert"https://www.zbmath.org/authors/?q=ai:burridge.robertA rezoning-free CESE scheme for solving the compressible Euler equations on moving unstructured meshes.https://www.zbmath.org/1453.761102021-02-27T13:50:00+00:00"Shen, Hua"https://www.zbmath.org/authors/?q=ai:shen.hua"Parsani, Matteo"https://www.zbmath.org/authors/?q=ai:parsani.matteoSummary: We construct a space-time conservation element and solution element (CESE) scheme for solving the compressible Euler equations on moving meshes (CESE-MM) which allow an arbitrary motion for each of the mesh points. The scheme is a direct extension of a purely Eulerian CESE scheme that was previously implemented on hybrid unstructured meshes [\textit{H. Shen} et al., ibid. 281, 375--402 (2015; Zbl 1354.65196)]. It adopts a staggered mesh in space and time such that the physical variables are continuous across the interfaces of the adjacent space-time control volumes and, therefore, a Riemann solver is not required to calculate interface fluxes or the node velocities. Moreover, the staggered mesh can significantly alleviate mesh tangles so that the time step can be kept at an acceptable level without using any rezoning operation. The discretization of the integral space-time conservation law is completely based on the physical space-time control volume, thereby satisfying the physical and geometrical conservation laws. Plenty of numerical examples are carried out to validate the accuracy and robustness of the CESE-MM scheme.A hybrid finite-volume, discontinuous Galerkin discretization for the radiative transport equation.https://www.zbmath.org/1453.820992021-02-27T13:50:00+00:00"Heningburg, Vincent"https://www.zbmath.org/authors/?q=ai:heningburg.vincent"Hauck, Cory D."https://www.zbmath.org/authors/?q=ai:hauck.cory-dA surface moving mesh method based on equidistribution and alignment.https://www.zbmath.org/1453.650432021-02-27T13:50:00+00:00"Kolasinski, Avary"https://www.zbmath.org/authors/?q=ai:kolasinski.avary"Huang, Weizhang"https://www.zbmath.org/authors/?q=ai:huang.weizhang.1|huang.weizhangSummary: A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed based on the current mesh. A selection of two- and three-dimensional examples are presented.Extending the stability limit of explicit scheme with spatial filtering for solving wave equations.https://www.zbmath.org/1453.652152021-02-27T13:50:00+00:00"Gao, Yingjie"https://www.zbmath.org/authors/?q=ai:gao.yingjie"Zhang, Jinhai"https://www.zbmath.org/authors/?q=ai:zhang.jinhai"Yao, Zhenxing"https://www.zbmath.org/authors/?q=ai:yao.zhenxingSummary: The explicit finite-difference scheme is widely used for solving the wave equation; however, its time step is strictly constrained by the Courant-Friedrichs-Lewy (CFL) stability limit. We apply spatial filtering to reduce the high-wavenumber noise caused by using a larger time step beyond the CFL stability limit. We further incorporate the forward and inverse time dispersion transforms to reduce the time-dispersion error. This method allows us to use a time step larger than the maximum time step allowed by the CFL stability limit.A space-time discontinuous Petrov-Galerkin method for acoustic waves.https://www.zbmath.org/1453.653162021-02-27T13:50:00+00:00"Ernesti, Johannes"https://www.zbmath.org/authors/?q=ai:ernesti.johannes"Wieners, Christian"https://www.zbmath.org/authors/?q=ai:wieners.christianSummary: We apply the discontinuous Petrov-Galerkin (DPG) method to linear acoustic waves in space and time using the framework of first-order Friedrichs systems. Based on results for operators and semigroups of hyperbolic systems, we show that the ideal DPG method is well-posed. The main task is to avoid the explicit use of traces, which are difficult to define in Hilbert spaces with respect to the graph norm of the space-time differential operator. Then, the practical DPG method is analyzed by constructing a Fortin operator numerically. For our numerical experiments, we introduce a simplified DPG method with discontinuous ansatz functions on the faces of the space-time skeleton, where the error is bounded by an equivalent conforming DPG method. Examples for a plane-wave configuration confirms the numerical analysis, and the computation of a diffraction pattern illustrates a first step to applications.
For the entire collection see [Zbl 1425.65008].Sparse identification of truncation errors.https://www.zbmath.org/1453.652812021-02-27T13:50:00+00:00"Thaler, Stephan"https://www.zbmath.org/authors/?q=ai:thaler.stephan"Paehler, Ludger"https://www.zbmath.org/authors/?q=ai:paehler.ludger"Adams, Nikolaus A."https://www.zbmath.org/authors/?q=ai:adams.nikolaus-aSummary: This work presents a data-driven approach to the identification of spatial and temporal truncation errors for linear and nonlinear discretization schemes of Partial Differential Equations (PDEs). Motivated by the central role of truncation errors, for example in the creation of implicit Large Eddy schemes, we introduce the \textit{Sparse Identification of Truncation Errors} (SITE) framework to automatically identify the terms of the modified differential equation from simulation data. We build on recent advances in the field of data-driven discovery and control of complex systems and combine it with classical work on modified differential equation analysis of Warming, Hyett, Lerat and Peyret. We augment a sparse regression-rooted approach with appropriate preconditioning routines to aid in the identification of the individual modified differential equation terms. The construction of such a custom algorithm pipeline allows attenuating of multicollinearity effects as well as automatic tuning of the sparse regression hyperparameters using the Bayesian information criterion (BIC). As proof of concept, we constrain the analysis to finite difference schemes and leave other numerical schemes open for future inquiry. Test cases include the linear advection equation with a forward-time, backward-space discretization, the Burgers' equation with a MacCormack predictor-corrector scheme and the Korteweg-de Vries equation with a Zabusky and Kruska discretization scheme. Based on variation studies, we derive guidelines for the selection of discretization parameters, preconditioning approaches and sparse regression algorithms. The results showcase highly accurate predictions underlining the promise of SITE for the analysis and optimization of discretization schemes, where analytic derivation of modified differential equations is infeasible.Space-time boundary element methods for the heat equation.https://www.zbmath.org/1453.652942021-02-27T13:50:00+00:00"Dohr, Stefan"https://www.zbmath.org/authors/?q=ai:dohr.stefan"Niino, Kazuki"https://www.zbmath.org/authors/?q=ai:niino.kazuki"Steinbach, Olaf"https://www.zbmath.org/authors/?q=ai:steinbach.olafSummary: In this chapter, we describe a space-time boundary element method for the numerical solution of the time-dependent heat equation. As model problem, we consider the initial Dirichlet boundary value problem, where the solution can be expressed in terms of given Dirichlet and initial data, and the unknown Neumann datum, which is determined by the solution of an appropriate boundary integral equation. For its numerical approximation, we consider a discretization, which is done with respect to a space-time decomposition of the boundary of the space-time domain. This space-time discretization technique allows us to parallelize the computation of the global solution of the whole space-time system. Besides the widely-used tensor product approach, we also consider an arbitrary decomposition of the spacetime boundary into boundary elements, allowing us to apply adaptive refinement in space and time simultaneously. In addition to the analysis of the boundary integral operators and the formulation of boundary element methods for the initial Dirichlet boundary value problem, we state a priori error estimates of the approximations. Moreover, we present numerical experiments to confirm the theoretical findings.
For the entire collection see [Zbl 1425.65008].Multiscale formulation of pore-scale compressible Darcy-Stokes flow.https://www.zbmath.org/1453.653042021-02-27T13:50:00+00:00"Guo, Bo"https://www.zbmath.org/authors/?q=ai:guo.bo"Mehmani, Yashar"https://www.zbmath.org/authors/?q=ai:mehmani.yashar"Tchelepi, Hamdi A."https://www.zbmath.org/authors/?q=ai:tchelepi.hamdi-aSummary: Direct numerical simulation (DNS) of fluid dynamics in digital images of porous materials is challenging due to the cut-off length issue where interstitial voids below the resolution of the imaging instrument cannot be resolved. Such subresolution microporosity can be critical for flow and transport because they could provide important flow pathways. A micro-continuum framework can be used to address this problem, which applies to the entire domain a single momentum equation, i.e., Darcy-Brinkman-Stokes (DBS) equation, that recovers Stokes equation in the resolved void space (i.e., macropores) and Darcy equation in the microporous regions. However, the DBS-based micro-continuum framework is computationally demanding. Here, we develop an efficient multiscale method for the compressible Darcy-Stokes flow arising from the micro-continuum approach. The method decomposes the domain into subdomains that either belong to the macropores or the microporous regions, on which Stokes or Darcy problems are solved locally, only once, to build basis functions. The nonlinearity from compressible flow is accounted for in a local correction problem on each subdomain. A global interface problem is solved to couple the local bases and correction functions to obtain an approximate global multiscale solution, which is in excellent agreement with the reference single-scale solution. The multiscale solution can be improved through an iterative strategy that guarantees convergence to the single-scale solution. The method is computationally efficient and well-suited for parallelization to simulate fluid dynamics in large high-resolution digital images of porous materials.Covering of high-dimensional cubes and quantization.https://www.zbmath.org/1453.901312021-02-27T13:50:00+00:00"Zhigljavsky, Anatoly"https://www.zbmath.org/authors/?q=ai:zhigljavsky.anatoly-a"Noonan, Jack"https://www.zbmath.org/authors/?q=ai:noonan.jackSummary: As the main problem, we consider covering of a \(d\)-dimensional cube by \(n\) balls with reasonably large \(d\) (10 or more) and reasonably small \(n\), like \(n = 100\) or \(n = 1000\). We do not require the full coverage but only 90\% or 95\% coverage. We establish that efficient covering schemes have several important properties which are not seen in small dimensions and in asymptotical considerations, for very large \(n\). One of these properties can be termed `do not try to cover the vertices' as the vertices of the cube and their close neighbourhoods are very hard to cover and for large \(d\) there are far too many of them. We clearly demonstrate that, contrary to a common belief, placing balls at points which form a low-discrepancy sequence in the cube, results in a very inefficient covering scheme. For a family of random coverings, we are able to provide very accurate approximations to the coverage probability. We then extend our results to the problems of coverage of a cube by smaller cubes and quantization, the latter being also referred to as facility location. Along with theoretical considerations and derivation of approximations, we provide results of a large-scale numerical investigation.Angular adaptivity with spherical harmonics for Boltzmann transport.https://www.zbmath.org/1453.654202021-02-27T13:50:00+00:00"Dargaville, S."https://www.zbmath.org/authors/?q=ai:dargaville.steven"Buchan, A. G."https://www.zbmath.org/authors/?q=ai:buchan.andrew-g"Smedley-Stevenson, R. P."https://www.zbmath.org/authors/?q=ai:smedley-stevenson.richard-p"Smith, P. N."https://www.zbmath.org/authors/?q=ai:smith.paul-n"Pain, C. C."https://www.zbmath.org/authors/?q=ai:pain.christopher-cSummary: This paper describes an angular adaptivity algorithm for Boltzmann transport applications which uses \(P_n\) and filtered \(P_n\) expansions, allowing for different expansion orders across space/energy. Our spatial discretisation is specifically designed to use less memory than competing DG schemes and also gives us direct access to the amount of stabilisation applied at each node. For filtered \(P_n\) expansions, we then use our adaptive process in combination with this net amount of stabilisation to compute a spatially dependent filter strength that does not depend on \textit{a priori} spatial information. This applies heavy filtering only where discontinuities are present, allowing the filtered \(P_n\) expansion to retain high-order convergence where possible. Regular and goal-based error metrics are shown and both the adapted \(P_n\) and adapted filtered \(P_n\) methods show significant reductions in DOFs and runtime. The adapted filtered \(P_n\) with our spatially dependent filter shows close to fixed iteration counts and up to high-order is even competitive with \(P^0\) discretisations in problems with heavy advection.Detecting troubled-cells on two-dimensional unstructured grids using a neural network.https://www.zbmath.org/1453.653012021-02-27T13:50:00+00:00"Ray, Deep"https://www.zbmath.org/authors/?q=ai:ray.deep"Hesthaven, Jan S."https://www.zbmath.org/authors/?q=ai:hesthaven.jan-sSummary: In a recent paper [ibid. 367, 166--191 (2018; Zbl 1415.65229)], we proposed a new type of troubled-cell indicator to detect discontinuities in the numerical solutions of one-dimensional conservation laws. This was achieved by suitably training an artificial neural network on canonical local solution structures for conservation laws. The proposed indicator was independent of problem-dependent parameters, giving it an advantage over existing limiter-based indicators. In the present paper, we extend this approach to train a similar network capable of detecting troubled-cells on two-dimensional unstructured grids. The proposed network has a smaller architecture compared to its one-dimensional predecessor, making it computationally efficient. Several numerical results are presented to demonstrate the performance of the new indicator.Enforcing constraints for interpolation and extrapolation in generative adversarial networks.https://www.zbmath.org/1453.681592021-02-27T13:50:00+00:00"Stinis, Panos"https://www.zbmath.org/authors/?q=ai:stinis.panos"Hagge, Tobias"https://www.zbmath.org/authors/?q=ai:hagge.tobias-j"Tartakovsky, Alexandre M."https://www.zbmath.org/authors/?q=ai:tartakovsky.alexandre-m"Yeung, Enoch"https://www.zbmath.org/authors/?q=ai:yeung.enochSummary: Generative Adversarial Networks (GANs) are becoming popular machine learning choices for training generators. At the same time there is a concerted effort in the machine learning community to expand the range of tasks in which learning can be applied as well as to utilize methods from other disciplines to accelerate learning. With this in mind, in the current work we suggest ways to enforce given constraints in the output of a GAN generator both for interpolation and extrapolation (prediction). For the case of dynamical systems, given a time series, we wish to train GAN generators that can be used to predict trajectories starting from a given initial condition. In this setting, the constraints can be in algebraic and/or differential form. Even though we are predominantly interested in the case of extrapolation, we will see that the tasks of interpolation and extrapolation are related. However, they need to be treated differently. For the case of interpolation, the incorporation of constraints is built into the training of the GAN. The incorporation of the constraints respects the primary game-theoretic setup of a GAN so it can be combined with existing algorithms. However, it can exacerbate the problem of instability during training that is well-known for GANs. We suggest adding small noise to the constraints as a simple remedy that has performed well in our numerical experiments. The case of extrapolation (prediction) is more involved. During training, the GAN generator learns to interpolate a noisy version of the data and we enforce the constraints. This approach has connections with model reduction that we can utilize to improve the efficiency and accuracy of the training. Depending on the form of the constraints, we may enforce them also during prediction through a projection step. We provide examples of linear and nonlinear systems of differential equations to illustrate the various constructions.On the estimation of artificial dissipation and dispersion errors in a generic partial differential equation.https://www.zbmath.org/1453.652792021-02-27T13:50:00+00:00"Castiglioni, Giacomo"https://www.zbmath.org/authors/?q=ai:castiglioni.giacomo"Sun, Guangrui"https://www.zbmath.org/authors/?q=ai:sun.guangrui"Domaradzki, J. Andrzej"https://www.zbmath.org/authors/?q=ai:domaradzki.j-andrzejSummary: A previously developed method, which allows for the estimation of the numerical dissipation through a kinetic energy balance equation averaged over a sub-domain, was applied with success to Navier-Stokes solvers for compressible and incompressible flows. In this work we show that the method can be generalized to other Partial Differential Equations (PDEs) and that for the linear advection equation it is in agreement with the modified equation analysis. Novelty of this work is the extension of the original method to the estimation of the dispersive error. The extension is based on a split of the residual of the kinetic energy balance equation that allows for the estimation of both dissipative and dispersive coefficients through a least squares regression. The procedure is validated on the linear advection equation for several numerical schemes for which dispersive and dissipative errors are known. When the new method is applied to linear or non-linear PDEs the estimates of the numerical dissipation obtained using the original method are recovered. The obtained rigorous results further support the previous heuristic method for estimating numerical errors in the course of simulations performed with arbitrary Navier-Stokes solvers.Bratu-like equation arising in electrospinning process: a Green's function fixed-point iteration approach.https://www.zbmath.org/1453.651882021-02-27T13:50:00+00:00"Kafri, H. Q."https://www.zbmath.org/authors/?q=ai:kafri.h-q"Khuri, S. A."https://www.zbmath.org/authors/?q=ai:khuri.suheil-a"Sayfy, A."https://www.zbmath.org/authors/?q=ai:sayfy.ali-m-sSummary: The aim of this article is to solve a Bratu-like nonlinear differential equation arising in electrospinning and vibration-electrospinning process that was recently introduced by \textit{H. Y. Liu} and \textit{P. Wang} [``A short remark on Bratu-like equation arising in electrospinningand vibration-electrospinning process'', Carbohydrate Polymers, ARBPOL-D-13-02650 (2014)]. Electro-spinning process has been associated to Bratu equation through thermo-electro-hydrodynamics balance equations. The proposed strategy is based on embedding Green's function into Picard's fixed-point iteration scheme. Validation of the novel algorithm is demonstrated through numerical examples which comprise certain selected values of the parameters that appear in the modelled equation.On stochastic Galerkin approximation of the nonlinear Boltzmann equation with uncertainty in the fluid regime.https://www.zbmath.org/1453.650192021-02-27T13:50:00+00:00"Hu, Jingwei"https://www.zbmath.org/authors/?q=ai:hu.jingwei"Jin, Shi"https://www.zbmath.org/authors/?q=ai:jin.shi"Shu, Ruiwen"https://www.zbmath.org/authors/?q=ai:shu.ruiwenSummary: The Boltzmann equation may contain uncertainties in initial/boundary data or collision kernel. To study the impact of these uncertainties, a stochastic Galerkin (sG) method was proposed in [the first two authors, ibid. 315, 150--168 (2016; Zbl 1349.82088)] and studied in the kinetic regime. When the system is close to the fluid regime (the Knudsen number is small), the method would become prohibitively expensive due to the stiff collision term. In this work, we develop efficient sG methods for the Boltzmann equation that work for a wide range of Knudsen numbers, and investigate, in particular, their behavior in the fluid regime.Adaptive boundary element methods for the computation of the electrostatic capacity on complex polyhedra.https://www.zbmath.org/1453.780122021-02-27T13:50:00+00:00"Betcke, Timo"https://www.zbmath.org/authors/?q=ai:betcke.timo"Haberl, Alexander"https://www.zbmath.org/authors/?q=ai:haberl.alexander"Praetorius, Dirk"https://www.zbmath.org/authors/?q=ai:praetorius.dirkSummary: The accurate computation of the electrostatic capacity of three dimensional objects is a fascinating benchmark problem with a long and rich history. In particular, the capacity of the unit cube has widely been studied, and recent advances allow to compute its capacity to more than ten digits of accuracy. However, the accurate computation of the capacity for general three dimensional polyhedra is still an open problem. In this paper, we propose a new algorithm based on a combination of ZZ-type \textit{a posteriori} error estimation and effective operator preconditioned boundary integral formulations to easily compute the capacity of complex three dimensional polyhedra to 5 digits and more. While this paper focuses on the capacity as a benchmark problem, it also discusses implementational issues of adaptive boundary element solvers, and we provide codes based on the boundary element package Bempp to make the underlying techniques accessible to a wide range of practical problems.A new efficient parameter estimation algorithm for high-dimensional complex nonlinear turbulent dynamical systems with partial observations.https://www.zbmath.org/1453.650082021-02-27T13:50:00+00:00"Chen, Nan"https://www.zbmath.org/authors/?q=ai:chen.nan"Majda, Andrew J."https://www.zbmath.org/authors/?q=ai:majda.andrew-jSummary: Parameter estimation for high-dimensional complex nonlinear turbulent dynamical systems with only partial observations is an important and practical issue. However, most of the existing parameter estimation algorithms are computationally expensive in the presence of a large number of state variables or parameters. In this article, a parameter estimation algorithm is developed for high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. This algorithm exploits the closed analytical form of the conditional statistics to recover the unobserved trajectories in an optimal and deterministic way, which facilitates the calculation of the likelihood function and circumvents the computationally expensive data augmentation approach in sampling the unobserved trajectories as widely used in the literature. Such an efficient method of recovering the unobserved trajectories is then incorporated into a standard Markov chain Monte Carlo (MCMC) algorithm to estimate parameters in complex dynamical system using only a short period of training data. Next, in light of the dynamical features, two effective strategies are developed and incorporated into the algorithm that facilitates the parameter estimation of many high-dimensional systems. The first strategy involves a judicious block decomposition of the state variables such that the original problem is divided into several subproblems coupled in a specific way that allows an extremely cheap parallel computation for the parameter estimation. The second strategy exploits statistical symmetry for a further reduction of the computational cost when the system is statistically homogeneous. The new parameter estimation algorithm is applied to a two-layer Lorenz 96 model with 80 state variables and 162 parameters and the model mimics the realistic features of atmosphere wave propagations and excitable media. The efficient algorithm results in an accurate estimation of the parameters, which further allows a skillful prediction by the model with estimated parameters. Other simple nonlinear models are also used to illustrate the features of the new algorithm.A point-mass particle method for the simulation of immiscible multiphase flows on an Eulerian grid.https://www.zbmath.org/1453.761712021-02-27T13:50:00+00:00"Wenzel, E. A."https://www.zbmath.org/authors/?q=ai:wenzel.e-a"Garrick, S. C."https://www.zbmath.org/authors/?q=ai:garrick.sean-cSummary: We present an Eulerian-Lagrangian approach for the modeling and simulation of immiscible multiphase flow systems. The Naiver-Stokes equations are solved on a traditional Eulerian grid while the fluid mass and phase information is discretized by Lagrangian particles. The method is novel because the particles move with a velocity that enforces consistency between the particle field density and the fluid density. The approach is advantageous in that (i) an arbitrary number of phases are easily represented, (ii) the particles remain well-distributed in space, even near merging and diverging characteristics, (iii) mass conservation is easily controlled, and (iv) the methodology is applicable to a wide range of Courant numbers. The governing equations are derived and a numerical method is presented that is applicable to incompressible flows. Performance is assessed via standard two-dimensional and three-dimensional phase transport tests as a function of both Eulerian grid resolution and Lagrangian particle resolution. Results show that the shape error converges with first-order with respect to increasing either Eulerian grid resolution or particle resolution, while mass conservation errors converge with the square root. The method is shown to successfully simulate expanding elliptical regions, stationary and oscillating droplets, a droplet in shear flow, a Rayleigh-Taylor instability, and the air blast atomization of a droplet.A high-order discontinuous Galerkin method for level set problems on polygonal meshes.https://www.zbmath.org/1453.653322021-02-27T13:50:00+00:00"Lipnikov, Konstantin"https://www.zbmath.org/authors/?q=ai:lipnikov.konstantin-n"Morgan, Nathaniel"https://www.zbmath.org/authors/?q=ai:morgan.nathaniel-rSummary: We propose and analyze discontinuous Galerkin schemes for solving level set equations on polygonal meshes. For linear equations, we assume that the velocity is given only on the cell surface. Velocity inside mesh cells is approximated using virtual element projectors on polynomial spaces. For nonlinear equations, we use the Taylor expansion to approximate the normalized solution gradient. We analyze the new schemes for a set of typical level set problems using square and polygonal meshes. The numerical results indicate great potential for using polygonal meshes in applications.Mathematical analysis of finite volume preserving scheme for nonlinear Smoluchowski equation.https://www.zbmath.org/1453.652602021-02-27T13:50:00+00:00"Singh, Mehakpreet"https://www.zbmath.org/authors/?q=ai:singh.mehakpreet"Matsoukas, Themis"https://www.zbmath.org/authors/?q=ai:matsoukas.themis"Walker, Gavin"https://www.zbmath.org/authors/?q=ai:walker.gavinSummary: This study presents the convergence analysis of the recently developed finite volume preserving scheme [\textit{L. Forestier-Coste} and \textit{S. Mancini}, SIAM J. Sci. Comput. 34, No. 6, B840--B860 (2012; Zbl 1259.82054)] for approximating a coalescence or Smoluchowski equation. The idea of the finite volume scheme is to preserve the total volume in the system by modifying the coalescence kernel using the notion of overlapping bins (cells). The consistency of the finite volume scheme is examined thoroughly in order to prove second-order convergence on uniform, non-uniform smooth and locally uniform grids independently of the aggregation kernel. The theoretical observations of order of convergence is verified using the experimental order of convergence for analytically tractable kernels.A semi-implicit and unconditionally stable approximation of the surface tension in two-phase fluids.https://www.zbmath.org/1453.652722021-02-27T13:50:00+00:00"Lee, Byungjoon"https://www.zbmath.org/authors/?q=ai:lee.byungjoon"Yoon, Gangjoon"https://www.zbmath.org/authors/?q=ai:yoon.gangjoon"Min, Chohong"https://www.zbmath.org/authors/?q=ai:min.chohongSummary: We focus on the energy conservation of two-phase fluids, where the change in kinetic energy is balanced by the gravitational potential and the surface energy related to surface tension. We introduce an unconditionally stable approximation in the sense that the total energy does not increase with any time step. An analysis is presented to prove the unconditional stability property of the scheme, and numerical results are given to confirming the analysis.Stochastic and coarse-grained two-dimensional modeling of directional particle movement.https://www.zbmath.org/1453.760162021-02-27T13:50:00+00:00"Ott, William"https://www.zbmath.org/authors/?q=ai:ott.william"Timofeyev, Ilya"https://www.zbmath.org/authors/?q=ai:timofeyev.ilya"Weber, Thomas"https://www.zbmath.org/authors/?q=ai:weber.thomas-w|weber.thomas-aSummary: We study the evolution of interacting groups of pedestrians in two-dimensional geometries. We introduce a microscopic stochastic model that includes floor fields modeling the global flow of individual groups as well as local interaction rules. From this microscopic model we derive an analytically-tractable system of conservation laws that governs the evolution of the macroscopic pedestrian densities. Numerical simulations show good agreement between the system of conservation laws and the microscopic model, though the latter is slightly more diffusive. We conclude by deriving second-order corrections to the system of conservation laws.Quasi Monte Carlo integration and kernel-based function approximation on Grassmannians.https://www.zbmath.org/1453.650072021-02-27T13:50:00+00:00"Breger, Anna"https://www.zbmath.org/authors/?q=ai:breger.anna"Ehler, Martin"https://www.zbmath.org/authors/?q=ai:ehler.martin"Gräf, Manuel"https://www.zbmath.org/authors/?q=ai:graf.manuelSummary: Numerical integration and function approximation on compact Riemannian manifolds based on eigenfunctions of the Laplace-Beltrami operator have been widely studied in the recent literature. The standard example in numerical experiments is the Euclidean sphere. Here, we derive numerically feasible expressions for the approximation schemes on the Grassmannian manifold, and we present the associated numerical experiments on the Grassmannian. Indeed, our experiments illustrate and match the corresponding theoretical results in the literature.
For the entire collection see [Zbl 1373.42002].Adaptive cell method with constitutive matrix correction for simulating physical field on a coarse grid.https://www.zbmath.org/1453.654252021-02-27T13:50:00+00:00"Chen, Feng"https://www.zbmath.org/authors/?q=ai:chen.feng.1|chen.feng"Wang, Jiawei"https://www.zbmath.org/authors/?q=ai:wang.jiawei"Ba, Can"https://www.zbmath.org/authors/?q=ai:ba.can"Bai, Xubin"https://www.zbmath.org/authors/?q=ai:bai.xubin"Dong, Tianyu"https://www.zbmath.org/authors/?q=ai:dong.tianyu"Ma, Xikui"https://www.zbmath.org/authors/?q=ai:ma.xikuiSummary: An adaptive cell method is proposed in order to simulate physical fields on a coarse grid with high accuracy. Compared with the conventional cell method, the advance of the proposed adaptive method originates from the so-called node weight correction procedure by using a patch-recovery technique, which corrects the local constitutive relations and in turn mitigates the error source that results from the discretization of constitutive equations. Furthermore, the proposed method has been extended to general curvilinear grids, in which the construction of the local constitutive matrix is derived by using differential geometry. Finally, we have demonstrated by theoretical analysis and numerical results that the proposed strategies can achieve a high-order convergence as well as high-efficiency.On the convergence rates of energy-stable finite-difference schemes.https://www.zbmath.org/1453.652332021-02-27T13:50:00+00:00"Svärd, Magnus"https://www.zbmath.org/authors/?q=ai:svard.magnus"Nordström, Jan"https://www.zbmath.org/authors/?q=ai:nordstrom.janSummary: We consider constant-coefficient initial-boundary value problems, with a first or second derivative in time and a highest spatial derivative of order \(q\), and their semi-discrete finite difference approximations. With an internal truncation error of order \(p \geq 1\), and a boundary error of order \(r \geq 0\), we prove that the convergence rate is: \(\min(p, r + q)\). The assumptions needed for these results to hold are: i) The continuous problem is linear and well-posed (with a smooth solution). ii) The numerical scheme is consistent, nullspace consistent, nullspace invariant, and energy stable. These assumptions are often satisfied for Summation-By-Parts schemes.Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems.https://www.zbmath.org/1453.650762021-02-27T13:50:00+00:00"Dopico, Froilán M."https://www.zbmath.org/authors/?q=ai:dopico.froilan-m"Marcaida, Silvia"https://www.zbmath.org/authors/?q=ai:marcaida.silvia"Quintana, María C."https://www.zbmath.org/authors/?q=ai:quintana.maria-c"Van Dooren, Paul"https://www.zbmath.org/authors/?q=ai:van-dooren.paul-mSummary: This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. This new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2020 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation.Approximation of the Boltzmann collision operator based on Hermite spectral method.https://www.zbmath.org/1453.653672021-02-27T13:50:00+00:00"Wang, Yanli"https://www.zbmath.org/authors/?q=ai:wang.yanli"Cai, Zhenning"https://www.zbmath.org/authors/?q=ai:cai.zhenningSummary: Based on the Hermite expansion of the distribution function, we introduce a Galerkin spectral method for the spatially homogeneous Boltzmann equation with the realistic inverse-power-law models. A practical algorithm is proposed to evaluate the coefficients in the spectral method with high accuracy, and these coefficients are also used to construct new computationally affordable collision models. Numerical experiments show that our method captures the low-order moments very efficiently.High order positivity-preserving discontinuous Galerkin schemes for radiative transfer equations on triangular meshes.https://www.zbmath.org/1453.653482021-02-27T13:50:00+00:00"Zhang, Min"https://www.zbmath.org/authors/?q=ai:zhang.min.2|zhang.min.7|zhang.min.4|zhang.min|zhang.min.5|zhang.min.6|zhang.min.3|zhang.min.1"Cheng, Juan"https://www.zbmath.org/authors/?q=ai:cheng.juan"Qiu, Jianxian"https://www.zbmath.org/authors/?q=ai:qiu.jianxianSummary: It is an important and challenging issue for the numerical solution of radiative transfer equations to maintain both high order accuracy and positivity. For the two-dimensional radiative transfer equations, \textit{D. Ling} et al. give a counterexample [J. Sci. Comput. 77, No. 3, 1801--1831 (2018; Zbl 1407.65196)] showing that unmodulated discontinuous Galerkin (DG) solver based either on the \(P^k\) or \(Q^k\) polynomial spaces could generate negative cell averages even if the inflow boundary value and the source term are both positive (and, for time dependent problems, also a nonnegative initial condition). Therefore the positivity-preserving frameworks in [\textit{X. Zhang} and \textit{C.-W. Shu}, J. Comput. Phys. 229, No. 23, 8918--8934 (2010; Zbl 1282.76128)] and [\textit{X. Zhang} et al., J. Sci. Comput. 50, No. 1, 29--62 (2012; Zbl 1247.65131)] which are based on the value of cell averages being positive cannot be directly used to obtain a high order conservative positivity-preserving DG scheme for the radiative transfer equations neither on rectangular meshes nor on triangular meshes. In [\textit{D. Yuan} et al., SIAM J. Sci. Comput. 38, No. 5, A2987--A3019 (2016; Zbl 1351.65105)], when the cell average of DG schemes is negative, a rotational positivity-preserving limiter is constructed which could keep high order accuracy and positivity in the one-dimensional radiative transfer equations with \(P^k\) polynomials and could be straightforwardly extended to two-dimensional radiative transfer equations on rectangular meshes with \(Q^k\) polynomials (tensor product polynomials). This paper presents an extension of the idea of the above mentioned one-dimensional rotational positivity-preserving limiter algorithm to two-dimensional high order positivity-preserving DG schemes for solving steady and unsteady radiative transfer equations on triangular meshes with \(P^k\) polynomials. The extension of this method is conceptually plausible but highly nontrivial. We first focus on finding a special quadrature rule on a triangle which should satisfy some conditions. The most important one is that the quadrature points can be arranged on several line segments, on which we can use the one-dimensional rotational positivity-preserving limiter. Since the number of the quadrature points is larger than the number of basis functions of \(P^k\) polynomial space, we determine a \(k\)-th polynomial by a \(L_2\)-norm Least Square subject to its cell average being equal to the weighted average of the values on the quadrature points after using the rotational positivity-preserving limiter. Since the weights used here are the quadrature weights which are positive, then the cell average of the modified polynomial is nonnegative. And the final modified polynomial can be obtained by using the two-dimensional scaling positivity-preserving limiter on the triangular element. We theoretically prove that our rotational positivity-preserving limiter on triangular meshes could keep both high order accuracy and positivity. It is relatively simple to implement, and also does not affect convergence to weak solutions. The numerical results validate the high order accuracy and the positivity-preserving properties of our schemes. The advantage of the triangular meshes on handling complex domain is also presented in our numerical examples.Energy-stable staggered schemes for the shallow water equations.https://www.zbmath.org/1453.653152021-02-27T13:50:00+00:00"Duran, Arnaud"https://www.zbmath.org/authors/?q=ai:duran.arnaud"Vila, Jean-Paul"https://www.zbmath.org/authors/?q=ai:vila.jean-paul"Baraille, Rémy"https://www.zbmath.org/authors/?q=ai:baraille.remySummary: We focus on the development and analysis of staggered schemes for the two-dimensional non-linear Shallow Water equations with varying bathymetry. Semi-implicit and fully explicit time-discretizations are proposed. Particular attention is paid on non-linear stability results, principally considered here through discrete energy dissipation arguments. To address such an issue, specific convective fluxes are employed, implying diffusive terms relying on the pressure gradient. In addition of providing an explicit control of the discrete energy budget, the method is shown to preserve motionless steady states as well as the positivity of the water height. These properties are still satisfied in a fully explicit context, provided an appropriate discretization of the pressure gradient. These stability results make the approach particularly robust and efficient, for both coastal flows and low-Froude number regimes. As a result, in addition of a great ease of implementation, the presented schemes meet the operational requirements attached to the simulation of large and small scale oceanic flows.A quasi-optimal non-overlapping domain decomposition method for two-dimensional time-harmonic elastic wave problems.https://www.zbmath.org/1453.740462021-02-27T13:50:00+00:00"Mattesi, V."https://www.zbmath.org/authors/?q=ai:mattesi.vanessa"Darbas, M."https://www.zbmath.org/authors/?q=ai:darbas.marion"Geuzaine, C."https://www.zbmath.org/authors/?q=ai:geuzaine.christophe-aSummary: This article presents the construction of a new non-overlapping domain decomposition method (DDM) for two-dimensional elastic scattering problems. The method relies on a high-order Transmission Boundary Condition (TBC) between sub-domains, which accurately approximates the exact Dirichlet-to-Neumann map. First, we explain the derivation of this new TBC in the context of a non-overlapping DDM. Next, a mode-by-mode convergence study for a model problem is presented, which shows the new method to be quasi-optimal, i.e. with an optimal convergence rate for evanescent modes and an improved convergence rate for the other modes compared to the standard low-order Lysmer-Kuhlemeyer TBC. Finally, the effectiveness of the new DDM is demonstrated in a finite element context by analyzing the behavior of the method on high-frequency elastodynamic simulations.Low-rank factorizations in data sparse hierarchical algorithms for preconditioning symmetric positive definite matrices.https://www.zbmath.org/1453.650612021-02-27T13:50:00+00:00"Agullo, Emmanuel"https://www.zbmath.org/authors/?q=ai:agullo.emmanuel"Darve, Eric"https://www.zbmath.org/authors/?q=ai:darve.eric"Giraud, Luc"https://www.zbmath.org/authors/?q=ai:giraud.luc"Harness, Yuval"https://www.zbmath.org/authors/?q=ai:harness.yuvalTaylor states in stellarators: a fast high-order boundary integral solver.https://www.zbmath.org/1453.762302021-02-27T13:50:00+00:00"Malhotra, Dhairya"https://www.zbmath.org/authors/?q=ai:malhotra.dhairya"Cerfon, Antoine"https://www.zbmath.org/authors/?q=ai:cerfon.antoine-j"Imbert-Gérard, Lise-Marie"https://www.zbmath.org/authors/?q=ai:imbert-gerard.lise-marie"O'Neil, Michael"https://www.zbmath.org/authors/?q=ai:oneil.michaelSummary: We present a boundary integral equation solver for computing Taylor relaxed states in non-axisymmetric solid and shell-like toroidal geometries. The computation of Taylor states in these geometries is a key element for the calculation of stepped pressure stellarator equilibria. The integral representation of the magnetic field in this work is based on the generalized Debye source formulation, and results in a well-conditioned second-kind boundary integral equation. The integral equation solver is based on a spectral discretization of the geometry and unknowns, and the computation of the associated weakly-singular integrals is performed with high-order quadrature based on a partition of unity. The resulting scheme for applying the integral operator is then coupled with an iterative solver and suitable preconditioners. Several numerical examples are provided to demonstrate the accuracy and efficiency of our method, and a direct comparison with the leading code in the field is reported.A massively-parallel, unstructured overset method to simulate moving bodies in turbulent flows.https://www.zbmath.org/1453.760582021-02-27T13:50:00+00:00"Horne, Wyatt James"https://www.zbmath.org/authors/?q=ai:horne.wyatt-james"Mahesh, Krishnan"https://www.zbmath.org/authors/?q=ai:mahesh.krishnanSummary: An unstructured overset method capable of performing direct numerical simulation (DNS) and large eddy simulation (LES) of many \((O(10^5))\) moving bodies, utilizing many computational cores \((O(10^5))\), in turbulent, incompressible fluid flow is presented. Unstructured meshes are attached to bodies and placed within a fixed background domain. Body meshes are allowed to arbitrarily overlap and move throughout the domain. Within each mesh a high resolution, unstructured, non-dissipative finite volume method is used to solve for the flow field. Boundary conditions for each mesh are provided by interpolation from flow solutions on overlapping meshes. When many unstructured meshes of different resolution overlap, care is required in the connection between the different flow solutions. An interpolant is created which seeks to preserve volume conservation of flow quantities between meshes regardless of overlapping mesh differences. An implicit fractional step method is used for time advancement, requiring the calculation of a predicted fluid velocity and corrector pressure field. For the predictor step, the resulting interpolation is directly introduced into the implicit equations for the predicted flow field. For the corrected pressure field, the continuity between meshes is weakly enforced using a penalty formulation. The pressure formulation is symmetric, positive-definite and non-singular resulting in a formulation which is readily solvable using traditional iterative matrix inversion techniques. An Arbitrary Euler-Lagrangian (ALE) method coupled to a 6 degrees of freedom rigid body equation system (6-DOF) is used for body motion. For rotation, a quaternion representation is used to solve Euler's equations of rigid body motion. A linear spring damper model, which uses geometry information readily available from the overset assembly process, is used for collisions. Validation of the method for canonical flow fields is presented including assessment of order of accuracy and kinetic energy conservation properties. Particle-resolved direct numerical simulation (PR-DNS) of single particles in various flow fields are presented for validation. PR-DNS results of 50,000 spherical particles freely moving within turbulent channel flow are shown as a demonstration of the method at full scale. LES results of a marine propeller under crashback conditions are shown to demonstrate the ability to simulate highly unsteady turbulent flows over complex, moving geometries.Conservative and entropy stable solid wall boundary conditions for the compressible Navier-Stokes equations: adiabatic wall and heat entropy transfer.https://www.zbmath.org/1453.760682021-02-27T13:50:00+00:00"Dalcin, Lisandro"https://www.zbmath.org/authors/?q=ai:dalcin.lisandro-d"Rojas, Diego"https://www.zbmath.org/authors/?q=ai:rojas.diego-alexander"Zampini, Stefano"https://www.zbmath.org/authors/?q=ai:zampini.stefano"Del Rey Fernández, David C."https://www.zbmath.org/authors/?q=ai:del-rey-fernandez.david-c"Carpenter, Mark H."https://www.zbmath.org/authors/?q=ai:carpenter.mark-h"Parsani, Matteo"https://www.zbmath.org/authors/?q=ai:parsani.matteoSummary: We present a novel technique for the imposition of non-linear entropy conservative and entropy stable solid wall boundary conditions for the compressible Navier-Stokes equations in the presence of an adiabatic wall, or a wall with a prescribed heat entropy flow. The procedure relies on the formalism and mimetic properties of diagonal-norm, summation-by-parts and simultaneous-approximation-term operators, and is a generalization of previous works on discontinuous interface coupling [the last author et al., ibid. 290, 132--138 (2015; Zbl 1349.76250)] and solid wall boundary conditions [the last author et al., ibid. 292, 88--113 (2015; Zbl 1349.76639)]. Using the method of lines, a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions are coupled with an entropy-conservative or entropy-stable discrete interior operator. The resulting estimate mimics the global entropy estimate obtained at the continuous level. The boundary data at the wall are weakly imposed using a penalty flux approach and a simultaneous-approximation-term technique for both the conservative variables and the gradient of the entropy variables. Discontinuous spectral collocation operators (mass lumped nodal discontinuous Galerkin operators), on high-order unstructured grids, are used for the purpose of demonstrating the robustness and efficacy of the new procedure for weakly enforcing boundary conditions. Numerical simulations confirm the non-linear stability of the proposed technique, with applications to three-dimensional subsonic and supersonic flows. The procedure described is compatible with any diagonal-norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction schemes.A Jacobian-free approximate Newton-Krylov startup strategy for RANS simulations.https://www.zbmath.org/1453.761172021-02-27T13:50:00+00:00"Yildirim, Anil"https://www.zbmath.org/authors/?q=ai:yildirim.anil"Kenway, Gaetan K. W."https://www.zbmath.org/authors/?q=ai:kenway.gaetan-k-w"Mader, Charles A."https://www.zbmath.org/authors/?q=ai:mader.charles-a"Martins, Joaquim R. R. A."https://www.zbmath.org/authors/?q=ai:martins.joaquim-r-r-aSummary: The favorable convergence rates of Newton-Krylov-based solution algorithms have increased their popularity for computational fluid dynamics applications. Unfortunately, these methods perform poorly during the initial stages of convergence, particularly for three-dimensional Reynolds-averaged Navier-Stokes simulations. Addressing this problem requires the use of a globalization method such as pseudo-transient continuation, along with an approximate Newton-Krylov startup stage. This class of methods marches the solution in pseudo-time with a matrix-based approximate Jacobian that has a lower bandwidth and better conditioning properties compared with the exact Jacobian. However, this matrix-based approach also has shortcomings, including the large cost of computing and storing the approximate Jacobian and its preconditioner, along with the need for an updated Jacobian for every nonlinear iteration. To rectify these shortcomings, we use approximate residual formulations in a Jacobian-free approximate Newton-Krylov algorithm. With the approximate Jacobian, we compute the vector products by using the approximate residual computations in a matrix-free manner while forming a preconditioner based on the matrix-based approximate Jacobian. This approach keeps the approximate Jacobian up to date and mitigates the cost of forming a matrix-based Jacobian and its preconditioner at each iteration by lagging the preconditioner between nonlinear iterations. We use varying levels of approximations with the matrix-free approach and thereby demonstrate the trade-off between rate of convergence and the cost of each nonlinear iteration. The proposed implementation uses only the exact and approximate residual formulations and can therefore be generalized with minimal additional implementation effort to a range of solvers and discretizations. The code is available under an open-source license.A completely explicit scheme of Cauchy problem in BSLM for solving the Navier-Stokes equations.https://www.zbmath.org/1453.760342021-02-27T13:50:00+00:00"Kim, Philsu"https://www.zbmath.org/authors/?q=ai:kim.philsu"Kim, Dojin"https://www.zbmath.org/authors/?q=ai:kim.dojin"Piao, Xiangfan"https://www.zbmath.org/authors/?q=ai:piao.xiangfan"Bak, Soyoon"https://www.zbmath.org/authors/?q=ai:bak.soyoonSummary: This paper presents a backward semi-Lagrangian method (BSLM) with third-order convergence in both time and space for solving incompressible Navier-Stokes equations. The third-order backward differentiation formula for the total time derivative and the projection method for the steady-state Stokes equation are used. A fourth-order difference scheme together with a local bi-cubic interpolation is used to solve the resulted two governing equations for the velocity and pressure. This paper mainly focuses on the development of an efficient scheme for solving the nonlinear Cauchy problem of the characteristic curve. We employ a modified linear multi-step method of the implicit-type based on the error correction strategy. A novel contribution of this paper is the design of a completely explicit formula for the three foot-points. The proposed method is superior to existing methods in terms of computational costs and accuracy, allowing the use of a large time step size. The fully explicit formula for the foot-points significantly improves the performance of a common BSLM for a wide range of practical applications.An efficient and accurate MPI-based parallel simulator for streamer discharges in three dimensions.https://www.zbmath.org/1453.652552021-02-27T13:50:00+00:00"Lin, Bo"https://www.zbmath.org/authors/?q=ai:lin.bo"Zhuang, Chijie"https://www.zbmath.org/authors/?q=ai:zhuang.chijie"Cai, Zhenning"https://www.zbmath.org/authors/?q=ai:cai.zhenning"Zeng, Rong"https://www.zbmath.org/authors/?q=ai:zeng.rong"Bao, Weizhu"https://www.zbmath.org/authors/?q=ai:bao.weizhuSummary: In this paper, we propose an efficient and accurate message-passing interface (MPI)-based parallel simulator for streamer discharges in three dimensions using the fluid model. First, we propose a new second-order semi-implicit scheme for the temporal discretization of the model that relaxes the dielectric relaxation time restriction. Moreover, it requires solving the Poisson-type equation only once at each time step, while the classical second-order explicit schemes typically need to do twice. Second, we introduce a geometric multigrid preconditioned FGMRES solver that dramatically improves the efficiency of solving the Poisson-type equation with either constant or variable coefficients. We show numerically that no more than 4 iterations are required for the Poisson solver to converge to a relative residual of \(10^{- 8}\) during streamer simulations; the FGMRES solver is much faster than R\&B SOR and other Krylov subspace solvers. Last but not least, all the methods are implemented using MPI. The parallel efficiency of the code and the fast algorithmic performances are demonstrated by a series of numerical experiments using up to 2560 cores on the Tianhe2-JK clusters. For applications, we study a double-headed streamer discharge as well as the interaction between two streamers, using up to 10.7 billion mesh cells.A conservative solver for surface-tension-driven multiphase flows on collocated unstructured grids.https://www.zbmath.org/1453.761162021-02-27T13:50:00+00:00"Xie, Bin"https://www.zbmath.org/authors/?q=ai:xie.bin"Jin, Peng"https://www.zbmath.org/authors/?q=ai:jin.peng"Nakayama, Hiroki"https://www.zbmath.org/authors/?q=ai:nakayama.hiroki"Liao, ShiJun"https://www.zbmath.org/authors/?q=ai:liao.shijun"Xiao, Feng"https://www.zbmath.org/authors/?q=ai:xiao.fengSummary: We developed a novel conservative model for simulating incompressible multiphase flows on collocated unstructured grids. In conventional schemes, the divergence-free condition is enforced by the numerical approximations based on the least-square approach which inherently cannot guarantee its conservative property for volume integrated average (VIA) of velocity. In this study, point values (PVs) at cell vertices are defined in addition to its VIAs for the pressure variable and a novel discretized formulation is devised for the pressure Poisson equation so as to enforce the VIA of velocity to satisfy the divergence-free condition. Considering the large density ratios in the vicinity of fluid interface, we have also put forward a non-oscillatory and less dissipative reconstruction scheme to improve the resolvability of discontinuous solutions. For the surface-tension dominated flow problems, a novel reinitialization scheme is proposed to transform the abruptly-varying volume fractions into a smooth-varying level-set function which significantly improves the solution accuracy of curvature estimation so as to suppress the spurious currents in presence of the unphysical oscillation. The resulting multiphase model that combines above numerical methods and techniques, therefore adequately assures divergence-free condition to preserve mass conservation, effectively controls the numerical oscillations and dissipations in the vicinity of moving interface and accurately resolves surface tension force with substantially suppressed spurious currents. Various numerical examples have been presented which demonstrate the excellent performance of the newly developed model to predict both fluid dynamics and interfacial deformations with high-fidelity.A unified Eulerian framework for multimaterial continuum mechanics.https://www.zbmath.org/1453.652522021-02-27T13:50:00+00:00"Jackson, Haran"https://www.zbmath.org/authors/?q=ai:jackson.haran"Nikiforakis, Nikos"https://www.zbmath.org/authors/?q=ai:nikiforakis.nikosSummary: A framework for simulating the interactions between multiple different continua is presented. Each constituent material is governed by the same set of equations, differing only in terms of their equations of state and strain dissipation functions. The interfaces between any combination of fluids, solids, and vacuum are handled by a new Riemann Ghost Fluid Method, which is agnostic to the type of material on either side (depending only on the desired boundary conditions). The Godunov-Peshkov-Romenski (GPR) model is used for modelling the continua (having recently been used to solve a range of problems involving Newtonian and non-Newtonian fluids, and elastic and elastoplastic solids), and this study represents a novel approach for handling multimaterial problems under this model. The resulting framework is simple, yet capable of accurately reproducing a wide range of different physical scenarios. It is demonstrated here to accurately reproduce analytical results for known Riemann problems, and to produce expected results in other cases, including some featuring heat conduction across interfaces, and impact-induced deformation and detonation of combustible materials. The framework thus has the potential to streamline development of simulation software for scenarios involving multiple materials and phases of matter, by reducing the number of different systems of equations that require solvers, and cutting down on the amount of theoretical work required to deal with the interfaces between materials.An extension of the Shannon wavelets for numerical solution of integro-differential equations.https://www.zbmath.org/1453.654462021-02-27T13:50:00+00:00"Attary, Maryam"https://www.zbmath.org/authors/?q=ai:attary.maryamSummary: In this work, an extension of the algebraic formulation of the Shannon wavelets for the numerical solution of a class of Volterra integro-differential equation is proposed. Our approach is based on the connection coefficients of the Shannon wavelet and collocation method for constructing the algebraic equivalent representation of the problem. Also, the Shannon approximation is applied to solve one type of nonlinear integral equation arising from chemical phenomenon. An analysis of error for the problem is given. The obtained numerical results show the accuracy of the presented method.
For the entire collection see [Zbl 1381.00029].Discontinuous Galerkin methods for multi-layer ocean modeling: viscosity and thin layers.https://www.zbmath.org/1453.653232021-02-27T13:50:00+00:00"Higdon, Robert L."https://www.zbmath.org/authors/?q=ai:higdon.robert-lSummary: The work described here is part of a continuing project to develop, analyze, and test some procedures for using discontinuous Galerkin (DG) numerical methods in multi-layer, isopycnic models of ocean circulation. The steps taken in the present paper include the following. (1) Develop an implementation of horizontal viscosity for usage in DG methods for multi-layer models. This step involves a formulation of the local DG method that can be used in the context of barotropic-baroclinic splitting, a widely-used approach to handling the multiple time scales in ocean circulation models. (2) Develop techniques that enable a layered model to exhibit thin layers without computational failures. Layers with negligible thickness can develop in situations that include coastal upwelling, outcropping of surfaces of constant density to the upper boundary of the fluid due to lateral variations in temperature, or intersections of density surfaces with bottom topography. For the sake of DG computations involving thin layers, this paper develops (i) implementations of wind stress, bottom stress, and interfacial shear stress that do not provoke spuriously large velocities, and (ii) a limiter that maintains nonnegative layer thicknesses in DG solutions. (3) Test the above techniques in numerical experiments involving model problems.Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations.https://www.zbmath.org/1453.654382021-02-27T13:50:00+00:00"Wimmer, Golo A."https://www.zbmath.org/authors/?q=ai:wimmer.golo-a"Cotter, Colin J."https://www.zbmath.org/authors/?q=ai:cotter.colin-john"Bauer, Werner"https://www.zbmath.org/authors/?q=ai:bauer.wernerSummary: We present an energy conserving space discretisation of the rotating shallow water equations using compatible finite elements. It is based on an energy and enstrophy conserving Hamiltonian formulation as described in [\textit{A. T. T. McRae} and \textit{C. J. Cotter}, ``Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements'', Q. J. R. Meteorol. Soc. 140, No. 684, 2223--2234 (2014; \url{doi:10.1002/qj.2291})], and extends it to include upwinding in the velocity and depth advection to increase stability. Upwinding for velocity in an energy conserving context was introduced for the incompressible Euler equations in [\textit{A. Natale} and \textit{C. J. Cotter}, IMA J. Numer. Anal. 38, No. 3, 1388--1419 (2018; Zbl 1408.65069)], while upwinding in the depth field in a Hamiltonian finite element context is newly described here. The energy conserving property is validated by coupling the spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved field development with respect to stability when upwinding in the depth field is included.Gaussian wave packet transform based numerical scheme for the semi-classical Schrödinger equation with random inputs.https://www.zbmath.org/1453.650202021-02-27T13:50:00+00:00"Jin, Shi"https://www.zbmath.org/authors/?q=ai:jin.shi"Liu, Liu"https://www.zbmath.org/authors/?q=ai:liu.liu"Russo, Giovanni"https://www.zbmath.org/authors/?q=ai:russo.giovanni.2|russo.giovanni"Zhou, Zhennan"https://www.zbmath.org/authors/?q=ai:zhou.zhennanSummary: In this work, we study the semi-classical limit of the Schrödinger equation with random inputs, and show that the semi-classical Schrödinger equation produces \(O(\varepsilon)\) oscillations in the random variable space. With the Gaussian wave packet transform, the original Schrödinger equation is mapped to an ordinary differential equation (ODE) system for the wave packet parameters coupled with a partial differential equation (PDE) for the quantity \(w\) in rescaled variables. Further, we show that the \(w\) equation does not produce \(\epsilon\) dependent oscillations, and thus it is more amenable for numerical simulations. We propose multi-level sampling strategy in implementing the Gaussian wave packet transform, where in the most costly part, i.e. simulating the \(w\) equation, it is sufficient to use \(\epsilon\) independent samples. We also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm, and hopefully shed light on possible future directions.A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces.https://www.zbmath.org/1453.654552021-02-27T13:50:00+00:00"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.12|liu.yang.18|liu.yang|liu.yang.22|liu.yang.20|liu.yang.3|liu.yang.8|liu.yang.7|liu.yang.4|liu.yang.15|liu.yang.2|liu.yang.14|liu.yang.16|liu.yang.23|liu.yang.17|liu.yang.6|liu.yang.1|liu.yang.13|liu.yang.5|liu.yang.10|liu.yang.11|liu.yang.21|liu.yang.9|liu.yang.19"Yang, Haizhao"https://www.zbmath.org/authors/?q=ai:yang.haizhaoSummary: This paper introduces a hierarchical interpolative decomposition butterfly-LU factorization (H-IDBF-LU) preconditioner for solving two-dimensional electric-field integral equations (EFIEs) in electromagnetic scattering problems of perfect electrically conducting objects with open surfaces. H-IDBF-LU leverages the interpolative decomposition butterfly factorization (IDBF) to compress dense blocks of the discretized EFIE operator to expedite its application; this compressed operator also serves as an approximate LU factorization of the EFIE operator leading to an efficient preconditioner in iterative solvers. Both the memory requirement and computational cost of the H-IDBF-LU solver scale as \(O(N \log^2 N)\) in one iteration; the total number of iterations required for a reasonably good accuracy scales as \(O(1)\) to \(O(\log^2 N)\) in all of our numerical tests. The efficacy and accuracy of the proposed preconditioned iterative solver are demonstrated via its application to a broad range of scatterers involving up to 100 million unknowns.Stochastic multiscale flux basis for Stokes-Darcy flows.https://www.zbmath.org/1453.761772021-02-27T13:50:00+00:00"Ambartsumyan, Ilona"https://www.zbmath.org/authors/?q=ai:ambartsumyan.ilona"Khattatov, Eldar"https://www.zbmath.org/authors/?q=ai:khattatov.eldar"Wang, ChangQing"https://www.zbmath.org/authors/?q=ai:wang.changqing"Yotov, Ivan"https://www.zbmath.org/authors/?q=ai:yotov.ivanSummary: Three algorithms are developed for uncertainty quantification in modeling coupled Stokes and Darcy flows. The porous media may consist of multiple regions with different properties. The permeability is modeled as a non-stationary stochastic variable, with its log represented as a sum of local Karhunen-Loève (KL) expansions. The problem is approximated by stochastic collocation on either tensor-product or sparse grids, coupled with a multiscale mortar mixed finite element method for the spatial discretization. A non-overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem, which is solved by an iterative solver, for each stochastic realization. In the traditional domain decomposition implementation, each subdomain solves a local Dirichlet or Neumann problem in every interface iteration. To reduce this cost, two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced. The basis consists of the local flux (or velocity trace) responses from each mortar degree of freedom. It is computed for each subdomain independently before the interface iteration begins. The use of the multiscale flux basis avoids the need for subdomain solves on each iteration. The deterministic basis is computed at each stochastic collocation and used only at this realization. The stochastic basis is formed by further looping over all local realizations of a subdomain's KL region before the stochastic collocation begins. It is reused over multiple realizations. Numerical tests are presented to illustrate the performance of the three algorithms, with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms.IMEX HDG-DG: a coupled implicit hybridized discontinuous Galerkin and explicit discontinuous Galerkin approach for shallow water systems.https://www.zbmath.org/1453.653282021-02-27T13:50:00+00:00"Kang, Shinhoo"https://www.zbmath.org/authors/?q=ai:kang.shinhoo"Giraldo, Francis X."https://www.zbmath.org/authors/?q=ai:giraldo.francis-x"Bui-Thanh, Tan"https://www.zbmath.org/authors/?q=ai:bui-thanh.tanSummary: We propose IMEX HDG-DG schemes for planar and spherical shallow water systems. Of interest is subcritical flow, where the speed of the gravity wave is faster than that of nonlinear advection. In order to simulate these flows efficiently, we split the governing system into a stiff part describing the gravity wave and a non-stiff part associated with nonlinear advection. The former is discretized implicitly with the HDG method while an explicit Runge-Kutta DG discretization is employed for the latter. The proposed IMEX HDG-DG framework: 1) facilitates high-order solutions both in time and space; 2) avoids overly small time-step sizes; 3) requires only one linear system solve per time stage; 4) relative to DG generates smaller and sparser linear systems while promoting further parallelism; and 5) suppresses the fast modes in the system with a large time-step size. Numerical results for various test cases demonstrate that our methods are beneficial for applications where non-stiff terms are accurately treated while stiff terms are less accurately handled.Taylor expansion based fast multipole method for 3-d Helmholtz equations in layered media.https://www.zbmath.org/1453.780132021-02-27T13:50:00+00:00"Wang, Bo"https://www.zbmath.org/authors/?q=ai:wang.bo.1"Chen, Duan"https://www.zbmath.org/authors/?q=ai:chen.duan"Zhang, Bo"https://www.zbmath.org/authors/?q=ai:zhang.bo.9|zhang.bo.3|zhang.bo.1|zhang.bo.4|zhang.bo.7|zhang.bo.6|zhang.bo.5|zhang.bo.2|zhang.bo.8|zhang.bo"Zhang, Wenzhong"https://www.zbmath.org/authors/?q=ai:zhang.wenzhong"Cho, Min Hyung"https://www.zbmath.org/authors/?q=ai:cho.minhyung"Cai, Wei"https://www.zbmath.org/authors/?q=ai:cai.weiSummary: In this paper, we develop a Taylor expansion (TE) based fast multipole method (FMM) for low frequency 3D Helmholtz Green's function in layered media. Two forms of Taylor expansions, with either non-symmetric or symmetric derivatives of layered media Green's functions, are used for the implementations of the proposed TE-FMM. In the implementation with non-symmetric derivatives, an algorithm based on discrete complex image approximations and recurrence formulas is shown to be very efficient and accurate in computing the high order derivatives. Meanwhile, the implementation based on symmetric derivatives is more robust and pre-computed tables for the high order derivatives in translation operators are used. Numerical tests in layered media have validated the accuracy and \(O(N)\) complexity of the proposed algorithms.A conservative diffuse interface method for two-phase flows with provable boundedness properties.https://www.zbmath.org/1453.761362021-02-27T13:50:00+00:00"Mirjalili, Shahab"https://www.zbmath.org/authors/?q=ai:mirjalili.shahab"Ivey, Christopher B."https://www.zbmath.org/authors/?q=ai:ivey.christopher-b"Mani, Ali"https://www.zbmath.org/authors/?q=ai:mani.aliSummary: Central finite difference schemes have long been avoided in the context of two-phase flows for the advection of the phase indicator function due to numerical overshoots and undershoots associated with their dispersion errors. We will show however, for an incompressible flow, in the context of a specific diffuse interface model, one can maintain the boundedness of the phase field while also taking advantage of the low cost and ease of implementation of central differences to construct a non-dissipative discretization scheme for the advective terms. This is made possible by combining the advection and reinitialization steps of a conservative level set scheme introduced by [\textit{E. Olsson} and \textit{G. Kreiss}, J. Comput. Phys. 210, No. 1, 225--246 (2005; Zbl 1154.76368)] to form a phase field equation similar to that of [\textit{P.-H. Chiu} and \textit{Y.-T. Lin}, J. Comput. Phys. 230, No. 1, 185--204 (2011; Zbl 1427.76201)]. Instead of resorting to specialized upwind methods as in these articles, we prove that the boundedness of the phase field is guaranteed for certain choices of the free parameters \(\varepsilon\) and \(\gamma )\) for a specific central difference scheme that we propose. The proposed discretely conservative and bounded phase field equation, which is free of any reinitialization or mass redistribution, possesses desirable properties that can be leveraged in the coupled finite difference discretization of the two-phase momentum equation. Additionally, as compared to the state-of-the-art conservative and bounded two-phase flow methods, the proposed method boasts competitive accuracy-vs-cost trade-off, small memory requirements, ease of implementation, providing a viable alternative for realistic two-phase flow calculations. Given that the proposed method uses cheap, local numerical stencils with uniform loading throughout the computational domain, its implementation is expected to achieve superior parallel scalability compared to the commonly employed two-phase flow methods such as level set and VOF.A low dissipation method to cure the grid-aligned shock instability.https://www.zbmath.org/1453.760922021-02-27T13:50:00+00:00"Fleischmann, Nico"https://www.zbmath.org/authors/?q=ai:fleischmann.nico"Adami, Stefan"https://www.zbmath.org/authors/?q=ai:adami.stefan"Hu, Xiangyu Y."https://www.zbmath.org/authors/?q=ai:hu.xiangyu-y"Adams, Nikolaus A."https://www.zbmath.org/authors/?q=ai:adams.nikolaus-aSummary: The grid-aligned shock instability prevents an accurate computation of high Mach number flows using low-dissipation shock-capturing methods. In particular one manifestation, the so-called carbuncle phenomenon, has been investigated by various different groups over the past decades. Nevertheless, the mechanism of this instability is still not fully understood and commonly is suppressed by the introduction of additional numerical dissipation. However, present approaches may either significantly deteriorate the resolution of complex flow evolutions or involve additional procedures to limit stabilization measures to the shock region. Instead of increasing the numerical dissipation, in this paper, we present an alternative approach that relates the problem to the low Mach number in transverse direction of the shock front. We show that the inadequate scaling of the acoustic dissipation in the low Mach number limit is the prime reason for the instability. Our approach is to increase the ``numerical'' Mach number locally whenever the advection dissipation is small compared to the acoustic dissipation. A very simple modification of the eigenvalue calculation in the Roe approximation leads to a scheme with \textit{less numerical dissipation than the original Roe flux} which prevents the grid-aligned shock instability. The simplicity of the modification allows for a detailed investigation of multidimensional effects. By showing that modifications in flow direction affect the shock stability in the transverse directions we confirm the multidimensional nature of the instability. The efficiency and robustness of the modified scheme is demonstrated for a wide range of test cases that are known to be particularly prone to the shock instability. Moreover, the modified flux also is successfully applied to multi-phase flows.Gradient-consistent enrichment of finite element spaces for the DNS of fluid-particle interaction.https://www.zbmath.org/1453.760962021-02-27T13:50:00+00:00"Höllbacher, Susanne"https://www.zbmath.org/authors/?q=ai:hollbacher.susanne"Wittum, Gabriel"https://www.zbmath.org/authors/?q=ai:wittum.gabrielSummary: We present gradient-consistent enriched finite element spaces for the simulation of free particles in a fluid. This involves forces being exchanged between the particles and the fluid at the interface. In an earlier work [ibid. 393, 186--213 (2019; Zbl 1452.76117)] we derived a monolithic scheme which includes the interaction forces into the Navier-Stokes equations by means of a fictitious domain like strategy. Due to an inexact approximation of the interface oscillations of the pressure along the interface were observed. In multiphase flows oscillations and spurious velocities are a common issue. The surface force term yields a jump in the pressure and therefore the oscillations are usually resolved by extending the spaces on cut elements in order to resolve the discontinuity. For the construction of the enriched spaces proposed in this paper we exploit the Petrov-Galerkin formulation of the vertex-centered finite volume method (PG-FVM), as already investigated in [loc. cit.]. From the perspective of the finite volume scheme we argue that wrong discrete normal directions at the interface are the origin of the oscillations. The new perspective of normal vectors suggests to look at \textit{gradients} rather than \textit{values} of the enriching shape functions. The crucial parameter of the enrichment functions therefore is the gradient of the shape functions and especially the one of the \textit{test} space. The distinguishing feature of our construction therefore is an enrichment that is based on the choice of shape functions with consistent gradients. These derivations finally yield a fitted scheme for the immersed interface. We further propose a strategy ensuring a well-conditioned system independent of the location of the interface. The enriched spaces can be used within any existing finite element discretization for the Navier-Stokes equation. Our numerical tests were conducted using the PG-FVM. We demonstrate that the enriched spaces are able to eliminate the oscillations.Energy conserving local discontinuous Galerkin methods for the improved Boussinesq equation.https://www.zbmath.org/1453.653342021-02-27T13:50:00+00:00"Li, Xiaole"https://www.zbmath.org/authors/?q=ai:li.xiaole"Sun, Weizhou"https://www.zbmath.org/authors/?q=ai:sun.weizhou"Xing, Yulong"https://www.zbmath.org/authors/?q=ai:xing.yulong"Chou, Ching-Shan"https://www.zbmath.org/authors/?q=ai:chou.ching-shanSummary: The Boussinesq-type equations describe the propagation of weakly non-linear long waves in shallow waters and are widely applied to model water waves in shallow seas and harbors. In this paper, we propose a high-order local discontinuous Galerkin method to solve the improved Boussinesq equation, coupled with both explicit leap-frog and implicit midpoint energy-conserving time discretization. The proposed full-discrete method can be shown to conserve the discrete versions of both mass and energy of the continuous solution. The error estimate with optimal order of convergence is provided for the semi-discrete method. Our numerical experiments confirm optimal rates of convergence as well as the mass and energy conserving property, and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property. A series of numerical experiments are provided to show that the proposed method has the capability to simulate the interaction between two solitary waves, single wave break-up and blow-up behavior well.A face-area-weighted `centroid' formula for finite-volume method that improves skewness and convergence on triangular grids.https://www.zbmath.org/1453.652582021-02-27T13:50:00+00:00"Nishikawa, Hiroaki"https://www.zbmath.org/authors/?q=ai:nishikawa.hiroakiSummary: This paper proposes a face-area-weighted `centroid' as a superior alternative to the geometric centroid for defining a local origin in a cell-centered finite-volume method on triangular grids. It is demonstrated theoretically and numerically that the face-area-weighted `centroid' can reduce grid skewness and improve iterative convergence for triangular grids. It is also shown that source terms do not have to be integrated over a cell and can be evaluated simply at the local origin without losing the design order of accuracy. Numerical results demonstrate that the face-area-weighted `centroid' improves iterative convergence of an implicit defect-correction second-order finite-volume solver for inviscid and viscous flow problems on regular and irregular triangular grids.Delta Voronoi smoothed particle hydrodynamics, \(\delta\)-VSPH.https://www.zbmath.org/1453.761602021-02-27T13:50:00+00:00"Fernández-Gutiérrez, David"https://www.zbmath.org/authors/?q=ai:fernandez-gutierrez.david"Zohdi, Tarek I."https://www.zbmath.org/authors/?q=ai:zohdi.tarek-iSummary: A Lagrangian scheme that combines Voronoi diagrams with smoothed particle hydrodynamics (SPH) for incompressible flows has been developed. Within the Voronoi tessellation, the Voronoi particle hydrodynamics (VPH) method is used, which is structurally similar to SPH. Two sub-domains are defined based on the proximity to the boundaries. The VPH formulation is used for particles close to solid boundaries, where SPH consistency and implementation of boundary conditions become problematic. Some overlapping of both sub-domains is allowed in order to provide a buffer zone to progressively transition from one method to the other. An explicit weakly compressible formulation for both sub-domains is used, with the diffusive term from the \(\delta\)-SPH correction extended to the VPH formulation. In addition, the density field is periodically re-initialized and a shifting algorithm is included to avoid excessive deformation of the Voronoi cells. Solid, free-surface, and inlet/outlet boundary conditions are considered. A linear damping term is used during the initialization process to mitigate possible inconsistencies from the user-defined initial conditions. The accuracy of the coupled scheme is discussed by means of a set of well-known verification benchmarks.A numerical method for an inverse problem for Helmholtz equation with separable wavenumber.https://www.zbmath.org/1453.653952021-02-27T13:50:00+00:00"Wang, Yuanlong"https://www.zbmath.org/authors/?q=ai:wang.yuanlong"Tadi, M."https://www.zbmath.org/authors/?q=ai:tadi.mohsenSummary: This paper is concerned with inverse evaluation of the wavenumber for a Helmholtz equation. It is assumed that the wavenumber is composed of a known uniform background and an unknown separable part. This function can closely model an unhealthy abnormality in a healthy domain. The algorithm assumes an initial guess for the unknown perturbation part and obtains corrections to the guessed value. Numerical results indicate that the algorithm can recover close estimates of the unknown wavenumber based on boundary measurements.3-d topology optimization of modulated and oriented periodic microstructures by the homogenization method.https://www.zbmath.org/1453.740722021-02-27T13:50:00+00:00"Geoffroy-Donders, Perle"https://www.zbmath.org/authors/?q=ai:geoffroy-donders.perle"Allaire, Grégoire"https://www.zbmath.org/authors/?q=ai:allaire.gregoire"Pantz, Olivier"https://www.zbmath.org/authors/?q=ai:pantz.olivierSummary: This paper is motivated by the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. Generalizing our previous work in 2-d we propose a method for topology optimization of structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented. This method is made of three steps. The first step amounts to compute the homogenized properties of an adequately chosen parametrized microstructure (here, a cubic lattice with varying bar thicknesses). The second step optimizes the homogenized formulation of the problem, which is a classical problem of parametric optimization. The third, and most delicate, step projects the optimal oriented microstructure at a desired length scale. Compared to the 2-d case where rotations are parametrized by a single angle, to which a conformality constraint can be applied, the 3-d case is more involved and requires new ingredients. In particular, the full rotation matrix is regularized (instead of just one angle in 2-d) and the projection map which deforms the square periodic lattice is computed component by component. Several numerical examples are presented for compliance minimization in 3-d.A mixed mimetic spectral element model of the 3D compressible Euler equations on the cubed sphere.https://www.zbmath.org/1453.860252021-02-27T13:50:00+00:00"Lee, D."https://www.zbmath.org/authors/?q=ai:lee.dongjin|lee.deokwoo|lee.dashin|lee.doheon|lee.deokjung|lee.dongin|lee.donghyuk|lee.dongjun|lee.darin|lee.dongbin|lee.donghwa|lee.donghun|lee.dongchan|lee.donghee|lee.do-won|lee.dongkon|lee.dongwoo|lee.deokjae|lee.donghi|lee.dongyoung|lee.dongseop|lee.dongryeol|lee.duhee|lee.dooyoung|lee.donggeun|lee.dami|lee.deoggyu|lee.dohyoung|lee.donghwan|lee.daeyeol|lee.donghyun|lee.donwoo|lee.dean-j|lee.ding-wen|lee.denis|lee.dohoon|lee.donggyu|lee.daiki|lee.dongmyun|lee.derrick|lee.dongchun|lee.dongwon|lee.deishin|lee.dongju|lee.deukhee|lee.dongkyoo|lee.donghyeon|lee.dongjae|lee.duncan|lee.delman|lee.donna|lee.dongsoo|lee.dan-a|lee.daewon|lee.doobum|li.tatsien|lee.danhyang|lee.doyoon|lee.dongik|lee.daero|lee.dong-ho|lee.daeseok|lee.dean|lee.dahjye|lee.daeho|lee.ding|lee.daeyup|lee.daeshik|lee.dongsun|lee.darren|lee.dongkyu|lee.darrick|lee.dongyeup|lee.dockjin|lee.dabeen|lee.donghoon|lee.dohyung|lee.donggill|lee.dongwook|lee.dongyun|lee.doohann|lee.donghak|lee.dongman"Palha, A."https://www.zbmath.org/authors/?q=ai:palha.arturSummary: A model of the three-dimensional rotating compressible Euler equations on the cubed sphere is presented. The model uses a mixed mimetic spectral element discretization which allows for the exact exchanges of kinetic, internal and potential energy via the compatibility properties of the chosen function spaces. A Strang carryover dimensional splitting procedure is used, with the horizontal dynamics solved explicitly and the vertical dynamics solved implicitly so as to avoid the CFL restriction of the vertical sound waves. The function spaces used to represent the horizontal dynamics are discontinuous across vertical element boundaries, such that each horizontal layer is solved independently so as to avoid the need to invert a global 3D mass matrix, while the function spaces used to represent the vertical dynamics are similarly discontinuous across horizontal element boundaries, allowing for the serial solution of the vertical dynamics independently for each horizontal element. The model is validated against standard test cases for baroclinic instability within an otherwise hydrostatically and geostrophically balanced atmosphere, and a non-hydrostatic gravity wave as driven by a temperature perturbation.An adaptive reduced basis ANOVA method for high-dimensional Bayesian inverse problems.https://www.zbmath.org/1453.625872021-02-27T13:50:00+00:00"Liao, Qifeng"https://www.zbmath.org/authors/?q=ai:liao.qifeng"Li, Jinglai"https://www.zbmath.org/authors/?q=ai:li.jinglaiSummary: In Bayesian inverse problems sampling the posterior distribution is often a challenging task when the underlying models are computationally intensive. To this end, surrogates or reduced models are often used to accelerate the computation. However, in many practical problems, the parameter of interest can be of high dimensionality, which renders standard model reduction techniques infeasible. In this paper, we present an approach that employs the ANOVA decomposition method to reduce the model with respect to the unknown parameters, and the reduced basis method to reduce the model with respect to the physical parameters. Moreover, we provide an adaptive scheme within the MCMC iterations, to perform the ANOVA decomposition with respect to the posterior distribution. With numerical examples, we demonstrate that the proposed model reduction method can significantly reduce the computational cost of Bayesian inverse problems, without sacrificing much accuracy.Construction of some accelerated methods for solving scalar stochastic differential equations.https://www.zbmath.org/1453.650232021-02-27T13:50:00+00:00"Soheili, Ali R."https://www.zbmath.org/authors/?q=ai:soheili.ali-reza"Soleymani, Fazlollah"https://www.zbmath.org/authors/?q=ai:soleymani.fazlollahSummary: The purpose of this short communication is to contribute in constructing some new solvers for the strong solution of stochastic ordinary differential equations. It is derived that new methods could be obtained as new variants of the high order scheme of \textit{E. Platen} [Lect. Notes Control Inf. Sci. 96, 187--193 (1987; Zbl 0635.60070)]. A simple technique to accelerate their speed will also be pointed out. Finally, numerical simulations are brought forward to show the efficiency of the derived methods.Mesh-independent streamline tracing.https://www.zbmath.org/1453.650382021-02-27T13:50:00+00:00"Batista, David"https://www.zbmath.org/authors/?q=ai:batista.davidSummary: Following the theory of R-functions, a mesh cell of any type can be implicitly described by one single operator constructed from several distance-like functions which, combined with the transfinite interpolation technique and Darcy's velocities specified at the grid nodes, allows us to compute a velocity vector everywhere in the domain, producing a globally continuous vector field with locally performed calculations. Then, knowing a particle's initial position and using these velocities, one can compute its trajectory, obtaining simple equations to determine the exit point of the particle from a given cell. This alternative approach differs from standard schemes in that there is little dependence on the Eulerian topology for tracing the paths and all computations can be made directly at the physical space for any kind of mesh. Numerical experiments show the applicability of our method for streamline modeling on problems with hybrid grids, domains with non-trivial inner structures, and highly variating flows, among others.A weighted meshfree collocation method for incompressible flows using radial basis functions.https://www.zbmath.org/1453.653642021-02-27T13:50:00+00:00"Wang, Lihua"https://www.zbmath.org/authors/?q=ai:wang.lihua"Qian, Zhihao"https://www.zbmath.org/authors/?q=ai:qian.zhihao"Zhou, Yueting"https://www.zbmath.org/authors/?q=ai:zhou.yueting"Peng, Yongbo"https://www.zbmath.org/authors/?q=ai:peng.yongboSummary: A weighted strong form collocation method using radial basis functions and explicit time integration is proposed to solve the incompressible Navier-Stokes equations. The velocities and pressure are solved directly at the same time step and the continuity equation is satisfied at each time step which improve the solution accuracy and stability. No artificial compressibility coefficient needs to be introduced for modeling the incompressible flows and no pressure oscillation arises in the numerical solutions. Optimal convergence can be achieved by imposing the derived proper weights on the boundaries and the continuity equation. Radial basis collocation method in a Lagrangian form is quite easy to capture the moving boundary or free surface in flow problems. Moreover, solid boundary conditions can be enforced directly and no special treatments are required. Further, critical time step for the explicit time integration is predicted in the stability analysis and the influences on the stability are evaluated. Numerical studies validate the high accuracy as well as good stability of the presented method.Second-order invariant domain preserving ALE approximation of hyperbolic systems.https://www.zbmath.org/1453.653202021-02-27T13:50:00+00:00"Guermond, Jean-Luc"https://www.zbmath.org/authors/?q=ai:guermond.jean-luc"Popov, Bojan"https://www.zbmath.org/authors/?q=ai:popov.boyan"Saavedra, Laura"https://www.zbmath.org/authors/?q=ai:saavedra.lauraSummary: In this paper we introduce an invariant domain preserving arbitrary Lagrangian Eulerian method for solving hyperbolic systems. The time stepping is explicit and the approximation in space is done with continuous finite elements. The method is second-order in space and made invariant domain preserving by using a newly introduced convex limiting technique.Generation of nested quadrature rules for generic weight functions via numerical optimization: application to sparse grids.https://www.zbmath.org/1453.650532021-02-27T13:50:00+00:00"Keshavarzzadeh, Vahid"https://www.zbmath.org/authors/?q=ai:keshavarzzadeh.vahid"Kirby, Robert M."https://www.zbmath.org/authors/?q=ai:kirby.robert-mike|kirby.robert-m-ii"Narayan, Akil"https://www.zbmath.org/authors/?q=ai:narayan.akil-cSummary: We present a numerical framework for computing nested quadrature rules for various weight functions. The well-known Kronrod method extends the Gauss-Legendre quadrature by adding new optimal nodes to the existing Gauss nodes for integration of higher order polynomials. Our numerical method generalizes the Kronrod rule for any continuous probability density function on real line with finite moments. We develop a bi-level optimization scheme to solve moment-matching conditions for two levels of main and nested rule and use a penalty method to enforce the constraints on the limits of the nodes and weights. We demonstrate our nested quadrature rule for probability measures on finite/infinite and symmetric/asymmetric supports. We generate Gauss-Kronrod-Patterson rules by slightly modifying our algorithm and present results associated with Chebyshev polynomials which are not reported elsewhere. We finally show the application of our nested rules in construction of sparse grids where we validate the accuracy and efficiency of such nested quadrature-based sparse grids on parameterized boundary and initial value problems in multiple dimensions.The conservative splitting domain decomposition method for multicomponent contamination flows in porous media.https://www.zbmath.org/1453.653052021-02-27T13:50:00+00:00"Liang, Dong"https://www.zbmath.org/authors/?q=ai:liang.dong"Zhou, Zhongguo"https://www.zbmath.org/authors/?q=ai:zhou.zhongguoSummary: In the paper, a new conservative splitting decomposition method (S-DDM) is developed for computing nonlinear multicomponent contamination flows in porous media over multi-block sub-domains. On each block-divided sub-domain, we take three steps to solve the coupled nonlinear system of water-head equation and multicomponent concentration equations in each time interval. The interface Darcy's velocity and the interface global concentration fluxes are first predicted by the semi-implicit flux schemes, while the solutions of water-head and multicomponent concentrations, Darcy's velocity and global concentration fluxes in the interiors of sub-domains are computed by one-directional splitting implicit solution-flux coupled schemes on staggered meshes, and finally the interface Darcy velocity and global concentration fluxes are corrected by the interior solutions. The significance of our scheme is that while it keeps the advantages of the non-overlapping domain decomposition and the splitting technique, it preserves mass on the whole domain of domain decompositions. Numerical experiments are presented to illustrate the excellent performance of our proposed conservative S-DDM approach for computing nonlinear multicomponent contamination flows in groundwater. The developed algorithm of the conservative S-DDM works efficiently over multiple block-divided sub-domains, which can be applied in simulation of large scale multicomponent contamination flows in parallel computing.A modification of the parameterization method for a linear boundary value problem for a Fredholm integro-differential equation.https://www.zbmath.org/1453.654502021-02-27T13:50:00+00:00"Dzhumabaev, D. S."https://www.zbmath.org/authors/?q=ai:dzhumabaev.dulat-syzdykbekovich"Nazarova, K. Zh."https://www.zbmath.org/authors/?q=ai:nazarova.k-zh"Uteshova, R. E."https://www.zbmath.org/authors/?q=ai:uteshova.roza-eSummary: A modification of the parameterization method is proposed to solve a linear two-point boundary value problem for a Fredholm integro-differential equation. The domain of the problem is partitioned and additional parameters are set as the values of the solution at interior points of the partition subintervals. Definition of a regular pair consisting of a partition and chosen interior points is given. The original problem is transformed into a multipoint boundary value problem with parameters. For fixed values of parameters, we get a special Cauchy problem for a system of integro-differential equations on the subintervals. Using the solution to this problem, the boundary condition and continuity conditions of solutions at the interior mesh points of the partition, we construct a system of linear algebraic equations in parameters. It is established that the solvability of the problem under consideration is equivalent to that of the constructed system.3D hybrid mesh generation with an improved vertical stretch algorithm for geometric models with pinch-out features.https://www.zbmath.org/1453.654262021-02-27T13:50:00+00:00"Sun, Lu"https://www.zbmath.org/authors/?q=ai:sun.lu"Zhao, Guoqun"https://www.zbmath.org/authors/?q=ai:zhao.guoqun"Yeh, Gour-Tsyh"https://www.zbmath.org/authors/?q=ai:yeh.gourtsyh|yeh.gour-tsyhSummary: Aiming at the infeasibility of using quadrilateral-prisms to describe sharp shapes, an improved vertical stretch algorithm was proposed to generate three-dimensional (3D) hybrid meshes based on the quadrilateral mesh converted from triangles to discretize the geometric models with pinch-out features. A robust and automatic mesh generator was developed, which served as a preprocessing tool for simulating metal forming, water flows and transports. An initial mixed mesh composed of cuspate hexahedra, quadrilateral- and triangular-prisms was constructed to capture the sharp features of pinch-out regions effectively. Five splitting templates were established to divide the 3D elements derived from pinch-out sub-domains into tetrahedra and triangular-prisms. The concept of splitting points was defined, based on which the appropriate implementation mode of each splitting template could be determined, so as to achieve the conformity and consistency of common faces and edges between different kinds of elements. Two topological optimization templates were established to improve the quality of degenerated triangular-prisms and tetrahedra. A new pinch-out boundary quadrilateral insertion method was proposed to avoid the generation of cuspate hexahedra, which simplified the conformity treatment strategies and improved the quality of the resulting mesh to a certain extent. Practical applications confirmed that the hybrid mesh generation methods proposed in this paper could discretize the geometric models with pinch-out features precisely. The requirements on mesh conformity and refinement were also able to be fully satisfied.Image-driven biophysical tumor growth model calibration.https://www.zbmath.org/1453.351142021-02-27T13:50:00+00:00"Scheufele, Klaudius"https://www.zbmath.org/authors/?q=ai:scheufele.klaudius"Subramanian, Shashank"https://www.zbmath.org/authors/?q=ai:subramanian.shashank"Mang, Andreas"https://www.zbmath.org/authors/?q=ai:mang.andreas"Biros, George"https://www.zbmath.org/authors/?q=ai:biros.george"Mehl, Miriam"https://www.zbmath.org/authors/?q=ai:mehl.miriamAn efficient numerical algorithm for solving viscosity contrast Cahn-Hilliard-Navier-Stokes system in porous media.https://www.zbmath.org/1453.760782021-02-27T13:50:00+00:00"Liu, Chen"https://www.zbmath.org/authors/?q=ai:liu.chen"Frank, Florian"https://www.zbmath.org/authors/?q=ai:frank.florian"Thiele, Christopher"https://www.zbmath.org/authors/?q=ai:thiele.christopher"Alpak, Faruk O."https://www.zbmath.org/authors/?q=ai:alpak.faruk-o"Berg, Steffen"https://www.zbmath.org/authors/?q=ai:berg.steffen"Chapman, Walter"https://www.zbmath.org/authors/?q=ai:chapman.walter-g"Riviere, Beatrice"https://www.zbmath.org/authors/?q=ai:riviere.beatrice-mSummary: Two-phase flow with viscosity contrast at the pore scale is modeled by a time-dependent Cahn-Hilliard-Navier-Stokes model and belongs to the class of diffuse interface method. The model allows for moving contact line and varying wettability. The numerical scheme utilizes an efficient pressure-correction projection algorithm, in conjunction with interior penalty discontinuous Galerkin schemes for space discretization developed within the framework of a distributed parallel pore-scale flow simulation system. The effect of viscosity contrast on the phase distribution is studied in relation with capillary forces and wettability. The algorithm is numerically robust and lends itself naturally to large-scale 3D numerical simulations.A new multi-population-based artificial bee colony for numerical optimisation.https://www.zbmath.org/1453.902242021-02-27T13:50:00+00:00"Yu, Gan"https://www.zbmath.org/authors/?q=ai:yu.ganSummary: This paper presents a new artificial bee colony (ABC) for numerical optimisation. The new approach called NABC, which employs a new multi-population strategy to enhance the diversity of population. The standard ABC is good at exploration, but poor at exploitation. To tackle this issue, a modified solution updating equation is utilised to generate new candidate solutions. Experiments are conducted on ten well-known benchmark functions. Results show that NABC achieves better solutions than the standard ABC and global best guided ABC.Random batch methods (RBM) for interacting particle systems.https://www.zbmath.org/1453.820652021-02-27T13:50:00+00:00"Jin, Shi"https://www.zbmath.org/authors/?q=ai:jin.shi.1|jin.shi"Li, Lei"https://www.zbmath.org/authors/?q=ai:li.lei.7|li.lei.3|li.lei.4|li.lei|li.lei.2|li.lei.1|li.lei.5|li.lei.6"Liu, Jian-Guo"https://www.zbmath.org/authors/?q=ai:liu.jian-guo.1|liu.jian-guoSummary: We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from \(O(N^2)\) per time step to \(O(N)\), for a system with \(N\) particles with binary interactions. On one hand, these methods are efficient Asymptotic-Preserving schemes for the underlying particle systems, allowing \(N\)-independent time steps and also capture, in the \(N\to\infty\) limit, the solution of the mean field limit which are nonlinear Fokker-Planck equations; on the other hand, the stochastic processes generated by the algorithms can also be regarded as new models for the underlying problems. For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems.Homotopy series solutions to time-space fractional coupled systems.https://www.zbmath.org/1453.653812021-02-27T13:50:00+00:00"Zhang, Jin"https://www.zbmath.org/authors/?q=ai:zhang.jin|zhang.jin.2|zhang.jin.3|zhang.jin.1"Cai, Ming"https://www.zbmath.org/authors/?q=ai:cai.ming"Chen, Bochao"https://www.zbmath.org/authors/?q=ai:chen.bochao"Wei, Hui"https://www.zbmath.org/authors/?q=ai:wei.huiSummary: We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.On three steps two-grid finite element methods for the 2D-transient Navier-Stokes equations.https://www.zbmath.org/1453.653082021-02-27T13:50:00+00:00"Bajpai, Saumya"https://www.zbmath.org/authors/?q=ai:bajpai.saumya"Pani, Amiya K."https://www.zbmath.org/authors/?q=ai:pani.amiya-kumarSummary: In this paper, an error analysis of a three steps two level Galerkin finite element method for the two dimensional transient Navier-Stokes equations is discussed. First of all, the problem is discretized in spatial direction by employing finite element method on a coarse mesh \(\mathcal{T}_{H}\) with mesh size \(H\). Then, in step two, the nonlinear system is linearized around the coarse grid solution, say, \(u_{H}\), which is similar to Newton's type iteration and the resulting linear system is solved on a finer mesh \(\mathcal{T}_{h}\) with mesh size \(h\). In step three, a correction is obtained through solving a linear problem on the finer mesh and an updated final solution is derived. Optimal error estimates in \(L^{\infty}(\mathbf{L}^{2})\)-norm, when \(h = \mathcal{O}(H^{2-\delta})\) and in \(L^\infty(\mathbf{H}^{1})\)-norm, when \(h = \mathcal{O}(H^{4-\delta})\) for the velocity and in \(L^{\infty}(L^{2})\)-norm, when \(h = \mathcal{O}(H^{4-\delta})\) for the pressure are established for arbitrarily small \(\delta > 0\). Further, under uniqueness assumption, these estimates are proved to be valid uniformly in time. Then, based on backward Euler method, a completely discrete scheme is analyzed and \textit{a priori} error estimates are derived. Results obtained in this paper are sharper than those derived earlier by two-grid methods. Finally, the paper is concluded with some numerical experiments.Interpolative decomposition butterfly factorization.https://www.zbmath.org/1453.650882021-02-27T13:50:00+00:00"Pang, Qiyuan"https://www.zbmath.org/authors/?q=ai:pang.qiyuan"Ho, Kenneth L."https://www.zbmath.org/authors/?q=ai:ho.kenneth-l"Yang, Haizhao"https://www.zbmath.org/authors/?q=ai:yang.haizhaoThe Gaussian quadrature after Gauss.https://www.zbmath.org/1453.010112021-02-27T13:50:00+00:00"Sanz-Serna, J. M."https://www.zbmath.org/authors/?q=ai:sanz-serna.jesus-mariaThis is a Spanish translation, summarized and provided with extensive comments, of the 1815 memoir \textit{Methodus nova integralium valores per approximationem inveniendi}, in which Gauss introduced the quadrature rules that bear his name. Gauss's approach differs significantly from what is presented nowadays as Gaussian quadrature in textbooks. The original memoir displays a mastery of work with series, in which the problem is rephrased as one of functional approximation, solved with the help of continuous fractions.
Reviewer: Victor V. Pambuccian (Glendale)The perturbation algorithm for the realization of a four-layer semi-discrete solution scheme of an abstract evolutionary problem.https://www.zbmath.org/1453.651192021-02-27T13:50:00+00:00"Rogava, Jemal"https://www.zbmath.org/authors/?q=ai:rogava.jemal-l"Gulua, David"https://www.zbmath.org/authors/?q=ai:gulua.davidSummary: In the present paper, we use the perturbation algorithm to reduce a purely implicit four-layer semi-discrete scheme for an abstract evolutionary equation to two-layer schemes. An approximate solution of the original problem is constructed using the solutions of these schemes. Estimates of the approximate solution error are proved in a Hilbert space.Adaptive non-intrusive reduced order modeling for compressible flows.https://www.zbmath.org/1453.761182021-02-27T13:50:00+00:00"Yu, Jian"https://www.zbmath.org/authors/?q=ai:yu.jian"Yan, Chao"https://www.zbmath.org/authors/?q=ai:yan.chao"Jiang, Zhenhua"https://www.zbmath.org/authors/?q=ai:jiang.zhenhua"Yuan, Wu"https://www.zbmath.org/authors/?q=ai:yuan.wu"Chen, Shusheng"https://www.zbmath.org/authors/?q=ai:chen.shushengSummary: An adaptive non-intrusive reduced basis (RB) method based on Gaussian process regression (GPR) is proposed for parametrized compressible flows. Adaptivity is pursued in the offline stage. The reduced basis by proper orthogonal decomposition (POD) is constructed iteratively to achieve a specified tolerance. For GPR, active data selection is used at each iteration, with standard deviation as the error indicator. To improve accuracy for shock-dominated flows, a properly designed simplified problem (SP) is considered as input of the regression models in addition to using parameters directly. Furthermore, a surrogate error model is constructed to serve as an efficient error estimator for the GPR models. Several two- and three-dimensional cases are conducted, including the inviscid nozzle flow, the inviscid NACA0012 airfoil flow and the inviscid M6 wing flow. For all the cases, the trained models are able to make efficient predictions with reasonable accuracy in the online stage. The SP-based approach is observed to result in biased sampling towards transonic regions. The regression models are further applied in sensitivity analysis, from which the solution of the two-dimensional cases are shown to be significantly more sensitive to input parameters than the wing flow. This is consistent to the comparison of convergence histories between the parameter-based and the SP-based models. For cases of high sensitivity, the SP-based approach is superior and can help to significantly reduce the number of required snapshots to achieve a prescribed tolerance.A cell edge-based linearity-preserving scheme for diffusion problems on two-dimensional unstructured grids.https://www.zbmath.org/1453.653902021-02-27T13:50:00+00:00"Luo, Longshan"https://www.zbmath.org/authors/?q=ai:luo.longshan"Gao, Zhiming"https://www.zbmath.org/authors/?q=ai:gao.zhiming"Wu, Jiming"https://www.zbmath.org/authors/?q=ai:wu.jimingSummary: We propose a new finite volume scheme for 2D anisotropic diffusion problems on general unstructured meshes. The main feature lies in the introduction of two auxiliary unknowns on each cell edge, and then the scheme has both cell-centered primary unknowns and cell edge-based auxiliary unknowns. The auxiliary unknowns are interpolated by the multipoint flux approximation technique, which reduces the scheme to a completely cell-centered one. The derivation of the scheme satisfies the linearity-preserving criterion that requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is then called as a cell edge-based linearity-preserving scheme. The optimal convergence rates are numerically obtained on unstructured grids in case that the diffusion tensor is taken to be anisotropic and/or discontinuous.Sparse approximation of fitting surface by elastic net.https://www.zbmath.org/1453.650312021-02-27T13:50:00+00:00"Hao, Yong-Xia"https://www.zbmath.org/authors/?q=ai:hao.yongxia"Lu, Dianchen"https://www.zbmath.org/authors/?q=ai:lu.dianchenSummary: The goal of this paper is to develop a computational model and corresponding efficient algorithm for obtaining a sparse representation of the fitting surface to the given scattered data. The basic idea of the model is to utilize the principal shift invariant space and the balanced \(l_{1}, l_{2}\) norm minimization (named elastic net). The elastic net can be solved efficiently by an adapted split Bregman iteration algorithm. Numerical experiments indicate that by choosing appropriate regularization parameters, the model can efficiently provide an acceptable compromise between the minimization of the data mismatch term and the sparsity of the solution.Identification of the source function for a seawater intrusion problem in unconfined aquifer.https://www.zbmath.org/1453.652922021-02-27T13:50:00+00:00"Slimani, S."https://www.zbmath.org/authors/?q=ai:slimani.said|slimani.souad|slimani.safia"Medarhri, I."https://www.zbmath.org/authors/?q=ai:medarhri.ibtissam"Najib, K."https://www.zbmath.org/authors/?q=ai:najib.khalid"Zine, A."https://www.zbmath.org/authors/?q=ai:zine.abdelmalekSummary: In this work, we study the inverse source problem of a seawater intrusion problem in an unconfined aquifer with sharp-diffuse interfaces. The model associated with the direct problem is nonlinear. We aim to reconstruct the source term following the technique used in [\textit{M. Kulbay} et al., Inverse Probl. Sci. Eng. 25, No. 2, 279--308 (2017; Zbl 1362.80005)]. As this technique is based on variable separation, a fixed-point strategy is adopted to linearize the problem. Numerical convergence is proven using some examples.Roots of quaternion polynomials: theory and computation.https://www.zbmath.org/1453.120042021-02-27T13:50:00+00:00"Sakkalis, Takis"https://www.zbmath.org/authors/?q=ai:sakkalis.takis"Ko, Kwanghee"https://www.zbmath.org/authors/?q=ai:ko.kwanghee"Song, Galam"https://www.zbmath.org/authors/?q=ai:song.galamSummary: A quaternion polynomial \(f(t)\) in the single variable \(t\), is one whose coefficients are in the skew field \(\mathbb{H}\) of quaternions. In this manuscript an elementary proof is given of the fact that such an \(f\) has a root in \(\mathbb{H}\). Moreover, an algorithm is proposed for finding all roots \(\zeta\) of \(f(t)\), along with their multiplicities. The algorithm is based on computing the real part of \(\zeta\) first, and then using the multiplication rule in \(\mathbb{H}\), the imaginary part of \(\zeta\) is computed via a linear quaternion equation. Several numerical examples are also presented to illustrate the performance of the method.A line search algorithm for wind field adjustment with incomplete data and RBF approximation.https://www.zbmath.org/1453.651402021-02-27T13:50:00+00:00"Cervantes, Daniel A."https://www.zbmath.org/authors/?q=ai:cervantes.daniel-a"González Casanova, Pedro"https://www.zbmath.org/authors/?q=ai:gonzalez-casanova.pedro"Gout, Christian"https://www.zbmath.org/authors/?q=ai:gout.christian"Moreles, Miguel Ángel"https://www.zbmath.org/authors/?q=ai:moreles.miguel-angelSummary: The problem of concern in this work is the construction of free divergence fields given scattered horizontal components. As customary, the problem is formulated as a PDE constrained least squares problem. The novelty of our approach is to construct the so-called adjusted field, as the unique solution along an appropriately chosen descent direction. The latter is obtained by the adjoint equation technique. It is shown that the classical adjusted field of Sasaki's is a particular case. On choosing descent directions, the underlying mass consistent model leads to the solution of an elliptic problem which is solved by means of a radial basis functions method. Finally, some numerical results for wind field adjustment are presented.Superconvergent Nyström method for Urysohn integral equations.https://www.zbmath.org/1453.654442021-02-27T13:50:00+00:00"Allouch, Chafik"https://www.zbmath.org/authors/?q=ai:allouch.chafik"Sbibih, Driss"https://www.zbmath.org/authors/?q=ai:sbibih.driss"Tahrichi, Mohamed"https://www.zbmath.org/authors/?q=ai:tahrichi.mohamedA superconvergent Nyström method is proposed for solving Urysohn nonlinear integral equations, i.e.,
\[
x(s) + \int\limits^1_0 k(s,t,x(t))dt = f(s), \qquad k, \frac{\partial u}{\partial x} \in C^{2r}, \quad r=1,2,\ldots .
\]
Using an interpolatory projection onto the set of \(r\) Gauss points, it is shown that the proposed method has an order of \(3r\) and one step of iteration improves the convergence order up to \(4r\). The size of the nonlinear system of equations that must be solved to calculate the approximate solution using this method remains the same as the range of the interpolatory projection. Numerical results are given to illustrate the improvement of the order.
Reviewer: Ilia V. Boikov (Penza)Semilocal convergence of a continuation method under \(\omega\)-differentiability condition.https://www.zbmath.org/1453.651222021-02-27T13:50:00+00:00"Prashanth, M."https://www.zbmath.org/authors/?q=ai:prashanth.maroju"Gupta, D. K."https://www.zbmath.org/authors/?q=ai:gupta.dharmendra-kumar"Motsa, S. S."https://www.zbmath.org/authors/?q=ai:motsa.sandile-sydneySummary: The aim of this paper is to study the semilocal convergence of a continuation method combining the Chebyshev's method and the convex acceleration of Newton's method for solving nonlinear operator equations in Banach spaces. This is carried out by deriving a family of recurrence relations based on two parameters under the assumption that the first Fréchet derivative satisfies the \(\omega\)-continuity condition given by \(\|F'(x) - F'(y)\| \leq \omega(\|x - y\|)\), \(x,y \in \Omega\), where \(\omega: \mathbb{R}^+ \to \mathbb{R}^+\) is a continuous and non-decreasing function such that \(\omega(0) \geq 0\). This condition generalises the Lipschitz and the Hölder continuity conditions on the first Fréchet derivative used for this purpose. Example can be given to show that the \(\omega\)-continuity condition works even when the Lipschitz and the Hölder continuity conditions on the first Fréchet derivative fail. This also avoids the computation of second Fréchet derivative which is either difficult to compute or unbounded at times. An existence and uniqueness theorem is established along with a priori error bounds. Two numerical examples are worked out to demonstrate the efficacy of our approach.Numerical approximation of multiple isolated roots of analytical systems.https://www.zbmath.org/1453.651072021-02-27T13:50:00+00:00"Giusti, Marc"https://www.zbmath.org/authors/?q=ai:giusti.marc"Yakoubsohn, Jean-Claude"https://www.zbmath.org/authors/?q=ai:yakoubsohn.jean-claudeSummary: The approximation of a multiple isolated root is a difficult problem. In fact the root can even be a repulsive root for a fixed point method like the Newton method. However there exists a huge literature on this topic but the answers given are not satisfactory. Numerical methods allowing a local convergence analysis work often under specific hypotheses. This viewpoint favouring numerical analysis forgets the geometry and the structure of the local algebra. Thus appeared so-called symbolic-numeric methods, yet full of lessons, but their precise numerical analysis is still missing. We propose in this paper a method of symbolic-numeric kind, whose numerical treatment is certified. The general idea is to construct a finite sequence of systems, admitting the same root, and called the \textit{deflation sequence}, so that the multiplicity of the root drops strictly between two successive systems. So the root becomes regular. Then we can extract a regular square we call \textit{deflated system}. We described already the construction of this deflated sequence when the singular root is known. The originality of this paper consists on one hand to construct a deflation sequence from a point close to the root and on the other hand to give a numerical analysis of this method. Analytic square integrable functions build the functional frame. Using the Bergman kernel, reproducing kernel of this functional frame, we are able to give a \textit{\(\alpha\)-theory à la Smale}. Furthermore we present new results on the determinacy of the numerical rank of a matrix and the closeness to zero of the evaluation map. As an important consequence we give an algorithm computing a deflation sequence \textit{free of \(\varepsilon\)}, threshold quantity measuring the numerical approximation, meaning that the entry of this algorithm does not involve the variable \(\varepsilon\).A transform based local RBF method for 2D linear PDE with Caputo-Fabrizio derivative.https://www.zbmath.org/1453.351782021-02-27T13:50:00+00:00"Kamran"https://www.zbmath.org/authors/?q=ai:kamran.muhammad-ahmad|kamran.farrukh|kamran.niloofar-n|kamran.fakhar|kamran.tayyab|kamran.niky|kamran.muhammad-sarwar|kamran.r|kamran.mohsin|kamran.kazem"Ali, Amjad"https://www.zbmath.org/authors/?q=ai:ali.amjad.1|ali.amjad"Gómez-Aguilar, José Francisco"https://www.zbmath.org/authors/?q=ai:gomez-aguilar.jose-franciscoSummary: The present work aims to approximate the solution of linear time fractional PDE with Caputo Fabrizio derivative. For the said purpose Laplace transform with local radial basis functions is used. The Laplace transform is applied to obtain the corresponding time independent equation in Laplace space and then the local RBFs are employed for spatial discretization. The solution is then represented as a contour integral in the complex space, which is approximated by trapezoidal rule with high accuracy. The application of Laplace transform avoids the time stepping procedure which commonly encounters the time instability issues. The convergence of the method is discussed also we have derived the bounds for the stability constant of the differentiation matrix of our proposed numerical scheme. The efficiency of the method is demonstrated with the help of numerical examples. For our numerical experiments we have selected three different domains, in the first test case the square domain is selected, for the second test the circular domain is considered, while for third case the L-shape domain is selected.Low-energy instruction precision assignment for multi-mode multiplier under accuracy and performance constraints.https://www.zbmath.org/1453.654682021-02-27T13:50:00+00:00"Kuang, S.-R."https://www.zbmath.org/authors/?q=ai:kuang.shiann-rong"Wu, K.-Y."https://www.zbmath.org/authors/?q=ai:wu.kaiya|wu.keyi|wu.kaiyu|wu.kaiyuan|wu.kuang-yao|wu.kuei-yang|wu.keying-y|wu.kuo-yeu|wu.kuen-yuh|wu.kaiyang|wu.kuo-yangSummary: Floating-point (FP) multipliers are the main energy consumers in many FP applications. Recently several FP multipliers with multiple-precision modes have been designed to trade energy consumption with output accuracy of FP multiplication operation (MOP). To effectively apply these multi-mode multipliers to FP applications, this paper presents a fast instruction precision assignment method for reducing energy consumption under accuracy and performance constraints. To easily set and check the accuracy constraint, we first build an affine arithmetic based error model to evaluate the overall output accuracy loss caused by inaccurate FP MOPs. Moreover, a simplified instruction scheduling method is also developed to quickly check the performance constraint. Based on these two check functions and the characteristics of proposed multi-mode multiplier, a fast Tabu search (TS) algorithm is then proposed to assign the precision mode of each FP MOP under the accuracy and performance constraints imposed on the given application. Experimental results show that the proposed fast TS algorithm can find the precision assignment with more energy saving and less searching time when compared to previous methods.Regularized DPSS preconditioners for generalized saddle point linear systems.https://www.zbmath.org/1453.650642021-02-27T13:50:00+00:00"Cao, Yang"https://www.zbmath.org/authors/?q=ai:cao.yang"Shi, Zhen-Quan"https://www.zbmath.org/authors/?q=ai:shi.zhen-quan"Shi, Quan"https://www.zbmath.org/authors/?q=ai:shi.quanSummary: By introducing a regularization matrix and an additional iteration parameter, a new class of regularized deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioners are proposed for generalized saddle point linear systems. Compared with the well-known Hermitian and skew-Hermitian splitting (HSS) preconditioner and the regularized HSS preconditioner [\textit{Z. Bai}, IMA J. Numer. Anal. 39, No. 4, 1888--1923 (2019; Zbl 07130811)] studied recently, the new RDPSS preconditioners have much better computing efficiency especially when the (1,1) block matrix is non-Hermitian. It is proved that the corresponding RDPSS stationary iteration method is unconditionally convergent. In addition, clustering property of the eigenvalues of the RDPSS preconditioned matrix is studied in detail. Two numerical experiments arising from the meshfree discretization of a static piezoelectric equation and the finite element discretization of the Navier-Stokes equation show the effectiveness of the new proposed preconditioners.Second order difference schemes for time-fractional KdV-Burgers' equation with initial singularity.https://www.zbmath.org/1453.652102021-02-27T13:50:00+00:00"Cen, Dakang"https://www.zbmath.org/authors/?q=ai:cen.dakang"Wang, Zhibo"https://www.zbmath.org/authors/?q=ai:wang.zhibo"Mo, Yan"https://www.zbmath.org/authors/?q=ai:mo.yanSummary: In this paper, we study the numerical method for time-fractional KdV-Burgers' equation with initial singularity. The famous \(L 2- 1_\sigma\) formula on graded meshes is adopted to approximate the Caputo derivative. Meanwhile, a nonlinear finite difference method on uniform grids is deduced for spatial discretization. The proposed method is second order in time and first order in space. With the help of the fractional Grönwall inequality, the unconditional stability and convergence of the current scheme are analyzed based on some skills. To raise the accuracy in spatial direction, a second order method is then carefully deduced. At last, theoretical results are verified by numerical experiments.Spectral approach to the scattering map for the semi-classical defocusing Davey-Stewartson II equation.https://www.zbmath.org/1453.370622021-02-27T13:50:00+00:00"Klein, Christian"https://www.zbmath.org/authors/?q=ai:klein.christian"McLaughlin, Ken"https://www.zbmath.org/authors/?q=ai:mclaughlin.kenneth-d-t-r"Stoilov, Nikola"https://www.zbmath.org/authors/?q=ai:stoilov.nikola-mSummary: The inverse scattering approach for the defocusing Davey-Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.On the hexagonal Shepard method.https://www.zbmath.org/1453.650302021-02-27T13:50:00+00:00"Dell'Accio, Francesco"https://www.zbmath.org/authors/?q=ai:dellaccio.francesco"Di Tommaso, Filomena"https://www.zbmath.org/authors/?q=ai:di-tommaso.filomenaThe authors consider the problem of scattered data interpolation on functions of two variables. More precisely, they introduce the so-called hexagonal Shepard method extending the Shepard and triangular Shepard methods to the case of six points. In doing so, the authors use the multinode basis functions [\textit{F. Dell'Accio} et al., Appl. Math. Comput. 318, 51--69 (2018; Zbl 1426.65012)] based on six points and local quadratic Lagrange polynomials that interpolate on the six points of each basis function. The global interpolant has quadratic precision and reaches cubic approximation order. Numerical experiments show performance of the new interpolation method.
Reviewer: Roberto Cavoretto (Torino)Sharp estimates for perturbation errors in summations.https://www.zbmath.org/1453.650972021-02-27T13:50:00+00:00"Lange, Marko"https://www.zbmath.org/authors/?q=ai:lange.marko"Rump, Siegfried M."https://www.zbmath.org/authors/?q=ai:rump.siegfried-michaelRecently the authors [BIT 57, No.\ 3, 927--941 (2017; Zbl 1380.65083)] obtained conditions under which classical bounds on roundoff errors in computing the sum of \(n\) numbers can be tightened. In this paper they derive tighter bounds based on almost arbitrary perturbations of real operations without any reference to a floating point grid. The bounds are sharp. In particular, for all feasible problem sizes, with IEEE 754 binary32 or binary64 format, there are examples satisfying the given bound with equality. The authors apply their results to derive error bounds for sums of products and for the multiplication of a vector by a Vandermonde matrix.
Reviewer: Alan L. Andrew (Melbourne)Numerical solution of a semi-linear inverse parabolic problem via HPM.https://www.zbmath.org/1453.653802021-02-27T13:50:00+00:00"Rostamian, Malihe"https://www.zbmath.org/authors/?q=ai:rostamian.malihe"Shahrezaee, Alimardan"https://www.zbmath.org/authors/?q=ai:shahrezaee.alimardanSummary: In this work, we consider the problem of finding an unknown time-dependent function in a semi-linear parabolic equation with given initial, boundary and energy over-specified conditions. The homotopy perturbation method, a powerful technique, is applied to obtain the numerical solution of this problem. Using this method, a rapid convergent sequence can be constructed which tends to the exact solution of the problem. Some examples are presented to illustrate the strength of the method.Simplicial partitions with applications to the finite element method.https://www.zbmath.org/1453.650032021-02-27T13:50:00+00:00"Brandts, Jan"https://www.zbmath.org/authors/?q=ai:brandts.jan-h"Korotov, Sergey"https://www.zbmath.org/authors/?q=ai:korotov.sergey"Křížek, Michal"https://www.zbmath.org/authors/?q=ai:krizek.michalPublisher's description: This monograph focuses on the mathematical and numerical analysis of simplicial partitions and the finite element method. This active area of research has become an essential part of physics and engineering, for example in the study of problems involving heat conduction, linear elasticity, semiconductors, Maxwell's equations, Einstein's equations and magnetic and gravitational fields.
These problems require the simulation of various phenomena and physical fields over complicated structures in three (and higher) dimensions. Since not all structures can be decomposed into simpler objects like $d$-dimensional rectangular blocks, simplicial partitions are important. In this book an emphasis is placed on angle conditions guaranteeing the convergence of the finite element method for elliptic PDEs with given boundary conditions.
It is aimed at a general mathematical audience who is assumed to be familiar with only a few basic results from linear algebra, geometry, and mathematical and numerical analysis.Classical Langevin dynamics derived from quantum mechanics.https://www.zbmath.org/1453.820642021-02-27T13:50:00+00:00"Hoel, Håkon"https://www.zbmath.org/authors/?q=ai:hoel.hakon-a"Szepessy, Anders"https://www.zbmath.org/authors/?q=ai:szepessy.andersSummary: The classical work by \textit{R. Zwanzig} [``Nonlinear generalized Langevin equations'', J. Stat. Phys. 9, 215--220 (1973; \url{doi:10.1007/BF01008729})] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.An enhanced bi-directional chaotic optimization algorithm.https://www.zbmath.org/1453.370912021-02-27T13:50:00+00:00"Derouiche, F."https://www.zbmath.org/authors/?q=ai:derouiche.f"Hamaizia, T."https://www.zbmath.org/authors/?q=ai:hamaizia.tayeb|hamaizia.taiebSummary: Based on the improved chaos searching strategy, an enhanced bi-directional chaotic optimization algorithm (EBCOA) is proposed in this study. A Lozi chaos mapping is used as a chaos generator to produce a chaos variable. In the process of EBCOA, and in order to make the chaos search more efficient, a new sub-step local chaos optimization method is proposed and a global search is done to find the current optimal solution in a certain range, and then a fine search reduces the space of optimized variables. Compared with the algorithm of traditional chaos search, the proposed algorithm is more accurate and can respond quickly. Simulation and experimental results confirm the efficiency of the proposed algorithm.Approximate solution of fractional Black-Scholes European option pricing equation by using ETHPM.https://www.zbmath.org/1453.911062021-02-27T13:50:00+00:00"Bhadane, Pradip R."https://www.zbmath.org/authors/?q=ai:bhadane.pradip-r"Ghadle, Kirtiwant P."https://www.zbmath.org/authors/?q=ai:ghadle.kirtiwant-p"Hamoud, Ahmed A."https://www.zbmath.org/authors/?q=ai:hamoud.ahmed-abdullahSummary: We proposed a new reliable combination of new Homotopy Perturbation Method (HPM) and Elzaki transform called as Elzaki Transform Homotopy Perturbation Method (ETHPM) is designed to obtain a exact solution to the fractional Black-Scholes equation with boundary condition for a European option pricing problem. The fractional derivative is in Caputo sense and the nonlinear terms in Fractional Black-Scholes Equation can be handled by using HPM. The Black-Scholes formula is used as a model for valuing European or American call and put options on a non-dividend paying stock. The methods give an analytic solution of the fractional Black-Scholes equation in the form of a convergent series. Finally, some examples are included to demonstrate the validity and applicability of the proposed technique.Computation of a contraction metric for a periodic orbit using meshfree collocation.https://www.zbmath.org/1453.340612021-02-27T13:50:00+00:00"Giesl, Peter"https://www.zbmath.org/authors/?q=ai:giesl.peterAuthor's abstract: Contraction analysis uses a local criterion to prove the long-term behavior of a dynamical system. We consider a contraction metric, i.e., a Riemannian metric with respect to which the distance between adjacent solutions contracts. If adjacent solutions in all directions perpendicular to the flow are contracted, then there exists a unique periodic orbit, which is exponentially stable. In this paper we propose a construction method using meshfree collocation to approximately solve a matrix-valued PDE problem. We derive error estimates and show that the approximation is itself a contraction metric if the collocation points are sufficiently dense. We apply the method to several examples.
Reviewer: Haiyan Wang (Phoenix)Parallel geometric multigrid.https://www.zbmath.org/1453.654302021-02-27T13:50:00+00:00"Martynenko, Sergey I."https://www.zbmath.org/authors/?q=ai:martynenko.sergey-i"Volokhov, Vadim M."https://www.zbmath.org/authors/?q=ai:volokhov.vadim-m"Yanovskiy, Leonid S."https://www.zbmath.org/authors/?q=ai:yanovskiy.leonid-sSummary: The paper describes practical approach for minimising the parallelisation overhead for a solution to the boundary value problems by geometric multigrid methods. It is shown that proposed multiple coarse grid correction strategy makes it possible not only to create the task of the smoother least demanding, but also to avoid load imbalance and limit the communication overhead. Estimation of maximum speedup and efficiency of parallel robust multigrid technique and parallel V-cycle are given.Hybrid Monte Carlo methods for sampling probability measures on submanifolds.https://www.zbmath.org/1453.650272021-02-27T13:50:00+00:00"Lelièvre, Tony"https://www.zbmath.org/authors/?q=ai:lelievre.tony"Rousset, Mathias"https://www.zbmath.org/authors/?q=ai:rousset.mathias"Stoltz, Gabriel"https://www.zbmath.org/authors/?q=ai:stoltz.gabrielSummary: Probability measures supported on submanifolds can be sampled by adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms. We show here how to generalize to GHMC a procedure suggested by Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds, where a reverse projection check is performed to enforce the reversibility of the algorithm for any timesteps and hence avoid biases in the invariant measure. We also provide a full mathematical analysis of such procedures, as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples.A fast solution method for time dependent multidimensional Schrödinger equations.https://www.zbmath.org/1453.650542021-02-27T13:50:00+00:00"Lanzara, F."https://www.zbmath.org/authors/?q=ai:lanzara.flavia"Maz'ya, V."https://www.zbmath.org/authors/?q=ai:mazya.vladimir-gilelevich"Schmidt, G."https://www.zbmath.org/authors/?q=ai:schmidt.gunther.2Summary: In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high-order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200.Bayesian optical flow with uncertainty quantification.https://www.zbmath.org/1453.624212021-02-27T13:50:00+00:00"Sun, Jie"https://www.zbmath.org/authors/?q=ai:sun.jie.2"Quevedo, Fernando J."https://www.zbmath.org/authors/?q=ai:quevedo.fernando-j"Bollt, Erik"https://www.zbmath.org/authors/?q=ai:bollt.erik-mA novel method based on augmented Markov vector process for the time-variant extreme value distribution of stochastic dynamical systems enforced by Poisson white noise.https://www.zbmath.org/1453.650112021-02-27T13:50:00+00:00"Lyu, Meng-Ze"https://www.zbmath.org/authors/?q=ai:lyu.meng-ze"Chen, Jian-Bing"https://www.zbmath.org/authors/?q=ai:chen.jianbing"Pirrotta, Antonina"https://www.zbmath.org/authors/?q=ai:pirrotta.antoninaSummary: The probability density function (PDF) of the time-variant extreme value process for structural responses is of great importance. Poisson white noise excitation occurs widely in practical engineering problems. The extreme value distribution of the response of systems excited by Poisson white noise processes is still not yet readily available. For this purpose, in the present paper, a novel method based on the augmented Markov vector process for the PDF of the time-variant extreme value process for a Poisson white noise driven dynamical system is proposed. Specifically, the augmented Markov vector (AMV) process is constructed by combining the extreme value process and its underlying response process. Then the joint probability density of the AMV can be evaluated by solving the Chapman-Kolmogorov Equation, e.g., via the path integral solution (PIS). Further, the PDF of the time-variant extreme value process is obtained, and can be used, say, to estimate the dynamic reliability of a stochastic system. For the purpose of illustration and verification, several numerical examples are studied and compared with Monte Carlo solution. Problems to be further studied are also discussed.Modelling uncertainties in phase-space boundary integral models of ray propagation.https://www.zbmath.org/1453.820612021-02-27T13:50:00+00:00"Bajars, Janis"https://www.zbmath.org/authors/?q=ai:bajars.janis"Chappell, David J."https://www.zbmath.org/authors/?q=ai:chappell.david-jSummary: A recently proposed phase-space boundary integral model for the stochastic propagation of ray densities is presented and, for the first time, explicit connections between this model and parametric uncertainties arising in the underlying physical model are derived. In particular, an asymptotic analysis for a weak noise perturbation of the propagation speed is used to derive expressions for the probability distribution of the phase-space boundary coordinates after transport along uncertain, and in general curved, ray trajectories. Furthermore, models are presented for incorporating geometric uncertainties in terms of both the location of an edge within a polygonal domain, as well as small scale geometric fluctuations giving rise to rough boundary reflections. Uncertain source terms are also considered in the form of stochastically distributed point sources and uncertain boundary data. A series of numerical experiments is then performed to illustrate these uncertainty models in two-dimensional convex polygonal domains.Multifidelity probability estimation via fusion of estimators.https://www.zbmath.org/1453.625132021-02-27T13:50:00+00:00"Kramer, Boris"https://www.zbmath.org/authors/?q=ai:kramer.boris"Marques, Alexandre Noll"https://www.zbmath.org/authors/?q=ai:marques.alexandre-noll"Peherstorfer, Benjamin"https://www.zbmath.org/authors/?q=ai:peherstorfer.benjamin"Villa, Umberto"https://www.zbmath.org/authors/?q=ai:villa.umberto-e"Willcox, Karen"https://www.zbmath.org/authors/?q=ai:willcox.karen-eSummary: This paper develops a multifidelity method that enables estimation of failure probabilities for expensive-to-evaluate models via information fusion and importance sampling. The presented general fusion method combines multiple probability estimators with the goal of variance reduction. We use low-fidelity models to derive biasing densities for importance sampling and then fuse the importance sampling estimators such that the fused multifidelity estimator is unbiased and has mean-squared error lower than or equal to that of any of the importance sampling estimators alone. By fusing all available estimators, the method circumvents the challenging problem of selecting the best biasing density and using only that density for sampling. A rigorous analysis shows that the fused estimator is optimal in the sense that it has minimal variance amongst all possible combinations of the estimators. The asymptotic behavior of the proposed method is demonstrated on a convection-diffusion-reaction partial differential equation model for which \(10^{5}\) samples can be afforded. To illustrate the proposed method at scale, we consider a model of a free plane jet and quantify how uncertainties at the flow inlet propagate to a quantity of interest related to turbulent mixing. Compared to an importance sampling estimator that uses the high-fidelity model alone, our multifidelity estimator reduces the required CPU time by 65\% while achieving a similar coefficient of variation.Existence results and iterative method for a fully fourth-order nonlinear integral boundary value problem.https://www.zbmath.org/1453.651702021-02-27T13:50:00+00:00"Dang, Quang A."https://www.zbmath.org/authors/?q=ai:dang-quang-a."Dang, Quang Long"https://www.zbmath.org/authors/?q=ai:dang.quang-longSummary: We consider the boundary value problem
\[
\begin{aligned}
u''''(t) &= f(t, u(t), u' (t), u''(t), u'''(t)), \quad 0 < t < 1, \\
u'(0) &= u''(0) = u'(1) = 0,\quad u(0) = \int\nolimits_0^1 g(s) u(s) ds,
\end{aligned}
\]
where \(f : [0, 1] \times \mathbb{R}^4 \rightarrow \mathbb{R}^+, a : [0, 1] \rightarrow \mathbb{R}^+\) are continuous functions. For \(f = f(u(t))\), very recently in [\textit{S. Benaicha} and \textit{F. Haddouchi}, ``Positive solutions of a nonlinear fourth-order integral boundary value problem'', An. Univ. Vest Timis. Ser. Mat.-Inform. 54, No. 1, 73--86 (2016; \url{doi:10.1515/awutm-2016-0005})] the existence of positive solutions was studied by employing the fixed point theory on cones. In this paper, by the method of reducing the boundary value problem to an operator equation for the right-hand sides we establish the existence, uniqueness, and positivity of solution and propose an iterative method on both continuous and discrete levels for finding the solution. We also give error analysis of the discrete approximate solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.Analysis and optimization of an adaptive interpolation algorithm for the numerical solution of a system of ordinary differential equations with interval parameters.https://www.zbmath.org/1453.651592021-02-27T13:50:00+00:00"Morozov, A. Yu."https://www.zbmath.org/authors/?q=ai:morozov.alexander-yu"Zhuravlev, A. A."https://www.zbmath.org/authors/?q=ai:zhuravlev.artem-a"Reviznikov, D. L."https://www.zbmath.org/authors/?q=ai:reviznikov.dmitry-lSummary: We consider an adaptive interpolation algorithm for numerical integration of systems of ordinary differential equations (ODEs) with interval parameters and initial conditions. At each time moment, a piecewise polynomial function of a prescribed degree is constructed in the course of algorithm operation that interpolates the dependence of the solution on particular values of interval uncertainties with a controlled accuracy. We study the question of computational costs of the algorithm. An analytic estimate is derived for the number of operations; it depends on the algorithm parameters, in particular, the degree of interpolation and the specific features of the ODE system being integrated. Using a number of representative examples of various dimension and containing a varying number of interval uncertainties, we show that there exists an optimum value of interpolation degree from the viewpoint of computational costs.A uniformly convergent numerical scheme for singularly perturbed differential equation with integral boundary condition arising in neural network.https://www.zbmath.org/1453.651822021-02-27T13:50:00+00:00"Shakti, D."https://www.zbmath.org/authors/?q=ai:shakti.d"Mohapatra, J."https://www.zbmath.org/authors/?q=ai:mohapatra.jugal|mohapatra.jeetSummary: This article deals with a singularly perturbed quasilinear boundary value problem with integral boundary condition which arises in neural network. The problem is discretised by using an upwind finite difference scheme on a non-uniform mesh obtained via equidistribution of a monitor function. We prove that the method is first order convergent in the discrete maximum norm independent of perturbation parameter. The parameter uniform convergence is confirmed by numerical computations.Estimation of a mixed effects model using a partially observed diffusion process.https://www.zbmath.org/1453.626202021-02-27T13:50:00+00:00"Soto, José"https://www.zbmath.org/authors/?q=ai:soto.jose"Infante, Saba"https://www.zbmath.org/authors/?q=ai:infante.saba"Camacho, Franklin"https://www.zbmath.org/authors/?q=ai:camacho.franklin"Amaro, Isidro R."https://www.zbmath.org/authors/?q=ai:amaro.isidro-rSummary: We consider a general mixed-effects model, where the variability of random effects of experimental individuals or units is incorporated through a stochastic differential equation. These models are useful for simultaneously analysing data from repeated measurements taken in discrete time and with errors. A Markov chain Monte Carlo algorithm was implemented to make the statistical inference a posteriori. A diagnostic analysis was carried out on the estimated parameters to detect if the model is suitable and show its convergence, in addition to the traces and posterior densities are shown. The methodology is illustrated using simulated data.Estimation of stochastic volatility models via auxiliary particles filter.https://www.zbmath.org/1453.627702021-02-27T13:50:00+00:00"Trosel, Yeniree"https://www.zbmath.org/authors/?q=ai:trosel.yeniree"Hernández, Aracelis"https://www.zbmath.org/authors/?q=ai:hernandez.aracelis"Infante, Saba"https://www.zbmath.org/authors/?q=ai:infante.sabaSummary: The growing interest in the study of volatility for series of financial instruments leads us to propose a methodology based on the versatility of the Sequential Monte Carlo (SMC) methods for the estimation of the states of the general stochastic volatility model (GSVM). In this paper, we proposed a methodology based on the state space structure applying filtering techniques such as the auxiliary particles filter for estimating the underlying volatility of the system. Additionally, we proposed to use a Markov chain Monte Carlo (MCMC) algorithm, such as is the Gibbs sampler for the estimation of the parameters. The methodology is illustrated through a series of returns of simulated data, and the series of returns corresponding to the Standard and Poor's 500 price index (S\&P 500) for the period 1995--2003. The results show that the proposed methodology allows to adequately explain the dynamics of volatility when there is an asymmetric response of this to a shock of a different sign, concluding that abrupt changes in returns correspond to high values in volatility.A blackbox polynomial system solver on parallel shared memory computers.https://www.zbmath.org/1453.654692021-02-27T13:50:00+00:00"Verschelde, Jan"https://www.zbmath.org/authors/?q=ai:verschelde.janSummary: A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods are applied to compute a numerical irreducible decomposition. Load balancing and pipelining are techniques in a parallel implementation on a computer with multicore processors. The application of the parallel algorithms is illustrated on solving the cyclic \(n\)-roots problems, in particular for \(n=8,9\), and \(12\).
For the entire collection see [Zbl 1396.68014].Generalized Mandelbrot and Julia sets in a family of planar angle-doubling maps.https://www.zbmath.org/1453.370402021-02-27T13:50:00+00:00"Hittmeyer, Stefanie"https://www.zbmath.org/authors/?q=ai:hittmeyer.stefanie"Krauskopf, Bernd"https://www.zbmath.org/authors/?q=ai:krauskopf.bernd"Osinga, Hinke M."https://www.zbmath.org/authors/?q=ai:osinga.hinke-mariaThe family of maps
\[f_{\lambda, c}=(1-\lambda+\lambda|z|^2)\left(\frac{z}{|z|}\right)^2+c\]
is here considered. When \(\lambda=1\) this family is the classic quadratic family \(f_c(z)=z^2+c\). The Julia set of \(f_{\lambda, c}\) is the boundary of the basin of \(\infty\) and the Mandelbrot set for a fixed \(\lambda\) is the set of \(c\) such that the orbit of the critical set is bounded under iteration.
When \(\lambda=1\), these sets have a similar properties as for the quadratic family \(f_c(z)=z^2+c\). For example, the Julia set is the closure of the repelling period cycles. The Mandelbrot set is the connected locus of the Julia set. However, when \(\lambda\in (0,1)\), they are not the same anymore. A lot of interesting phenomena happen and are fully discussed.
For the entire collection see [Zbl 1443.39002].
Reviewer: Tao Chen (New York)Convergence of Markovian stochastic approximation for Markov random fields with hidden variables.https://www.zbmath.org/1453.626192021-02-27T13:50:00+00:00"Qi, Anna"https://www.zbmath.org/authors/?q=ai:qi.anna"Yang, Lihua"https://www.zbmath.org/authors/?q=ai:yang.lihua"Huang, Chao"https://www.zbmath.org/authors/?q=ai:huang.chaoSymbolic-numerical algorithms for solving elliptic boundary-value problems using multivariate simplex Lagrange elements.https://www.zbmath.org/1453.654062021-02-27T13:50:00+00:00"Gusev, A. A."https://www.zbmath.org/authors/?q=ai:gusev.alexander-a"Gerdt, V. P."https://www.zbmath.org/authors/?q=ai:gerdt.vladimir-p"Chuluunbaatar, O."https://www.zbmath.org/authors/?q=ai:chuluunbaatar.ochbadrakh"Chuluunbaatar, G."https://www.zbmath.org/authors/?q=ai:chuluunbaatar.g"Vinitsky, S. I."https://www.zbmath.org/authors/?q=ai:vinitsky.sergue-i"Derbov, V. L."https://www.zbmath.org/authors/?q=ai:derbov.vladimir-l"Góźdź, A."https://www.zbmath.org/authors/?q=ai:gozdz.andrzej"Krassovitskiy, P. M."https://www.zbmath.org/authors/?q=ai:krassovitskiy.p-mSummary: We propose new symbolic-numerical algorithms implemented in Maple-Fortran environment for solving the self-adjoint elliptic boundary-value problem in a \(d\)-dimensional polyhedral finite domain, using the high-accuracy finite element method with multivariate Lagrange elements in the simplexes. The high-order fully symmetric PI-type Gaussian quadratures with positive weights and no points outside the simplex are calculated by means of the new symbolic-numerical algorithms implemented in Maple. Quadrature rules up to order 8 on the simplexes with dimension \(d=3\text{-}6\) are presented. We demonstrate the efficiency of algorithms and programs by benchmark calculations of a low part of spectra of exactly solvable Helmholtz problems for a cube and a hypercube.
For the entire collection see [Zbl 1396.68014].Analytical and numerical methods for solving hypersingular integral equations.https://www.zbmath.org/1453.654482021-02-27T13:50:00+00:00"Boykov, I. V."https://www.zbmath.org/authors/?q=ai:boykov.ilya-vSummary: The article gives a brief overview of works devoted to analytical and numerical methods for solving hypersingular integral equations. Hypersingular integral equations on closed and open integration loops, polyhypersingular and multidimensional hypersingular integral equations are considered. The main attention is paid to approximate methods for solving one-dimensional hypersingular integral equations of the first and second kinds with singularities of the second order. We consider hypersingular integral equations of the first kind defined on the segment \([-1,1]\), whose solutions have the form \(x(t)=(1-t^2)^{\pm 1/2} \varphi(t)\), \(((1-t)/(1+t))^{\pm 1/2}\varphi(t)\), where \(\varphi(t)\) is a smooth function. Hypersingular integral equations of the second kind, defined on the segment \([-1,1]\), are considered. For solution of its the spline-collocation method with first-order splines is constructed. A separate section is devoted to approximate methods for solving hypersingular integral equations on closed circuits.A continuation method for visualizing planar real algebraic curves with singularities.https://www.zbmath.org/1453.650402021-02-27T13:50:00+00:00"Chen, Changbo"https://www.zbmath.org/authors/?q=ai:chen.changbo"Wu, Wenyuan"https://www.zbmath.org/authors/?q=ai:wu.wenyuanSummary: We present a new method for visualizing planar real algebraic curves inside a bounding box based on numerical continuation and critical point methods. Since the topology of the curve near a singular point is not numerically stable, we trace the curve only outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is \(\epsilon\)-close to the curve. Such an approximation is more stable for defining the numerical connectedness of the complement of the curve, which is important for applications such as solving bi-parametric polynomial systems.{
}The algorithm starts by computing three types of key points of the curve, namely the intersection of the curve with small circles centered at singular points, regular critical points of every connected component of the curve, as well as intersection points of the curve with the given bounding box. It then traces the curve starting with and in the order of the above three types of points. This basic scheme is further enhanced by several optimizations, such as grouping singular points in natural clusters and tracing the curve by a try-and-resume strategy. The effectiveness of the algorithm is illustrated by numerous examples.
For the entire collection see [Zbl 1396.68014].Symbolic-numeric methods for nonlinear integro-differential modeling.https://www.zbmath.org/1453.651632021-02-27T13:50:00+00:00"Boulier, François"https://www.zbmath.org/authors/?q=ai:boulier.francois"Castel, Hélène"https://www.zbmath.org/authors/?q=ai:castel.helene"Corson, Nathalie"https://www.zbmath.org/authors/?q=ai:corson.nathalie"Lanza, Valentina"https://www.zbmath.org/authors/?q=ai:lanza.valentina"Lemaire, François"https://www.zbmath.org/authors/?q=ai:lemaire.francois"Poteaux, Adrien"https://www.zbmath.org/authors/?q=ai:poteaux.adrien"Quadrat, Alban"https://www.zbmath.org/authors/?q=ai:quadrat.alban"Verdière, Nathalie"https://www.zbmath.org/authors/?q=ai:verdiere.nathalieSummary: This paper presents a proof of concept for symbolic and numeric methods dedicated to the parameter estimation problem for models formulated by means of nonlinear integro-differential equations (IDE). In particular, we address: the computation of the model input-output equation and the numerical integration of IDE systems.
For the entire collection see [Zbl 1396.68014].A strongly consistent finite difference scheme for steady Stokes flow and its modified equations.https://www.zbmath.org/1453.761222021-02-27T13:50:00+00:00"Blinkov, Yury A."https://www.zbmath.org/authors/?q=ai:blinkov.yury-a"Gerdt, Vladimir P."https://www.zbmath.org/authors/?q=ai:gerdt.vladimir-p"Lyakhov, Dmitry A."https://www.zbmath.org/authors/?q=ai:lyakhov.dmitry-a"Michels, Dominik L."https://www.zbmath.org/authors/?q=ai:michels.dominik-ludewigSummary: We construct and analyze a strongly consistent second-order finite difference scheme for the steady two-dimensional Stokes flow. The pressure Poisson equation is explicitly incorporated into the scheme. Our approach suggested by the first two authors is based on a combination of the finite volume method, difference elimination, and numerical integration. We make use of the techniques of the differential and difference Janet/Gröbner bases. In order to prove strong consistency of the generated scheme we correlate the differential ideal generated by the polynomials in the Stokes equations with the difference ideal generated by the polynomials in the constructed difference scheme. Additionally, we compute the modified differential system of the obtained scheme and analyze the scheme's accuracy and strong consistency by considering this system. An evaluation of our scheme against the established marker-and-cell method is carried out.
For the entire collection see [Zbl 1396.68014].Sparse polynomial arithmetic with the BPAS library.https://www.zbmath.org/1453.654702021-02-27T13:50:00+00:00"Asadi, Mohammadali"https://www.zbmath.org/authors/?q=ai:asadi.mohammadali"Brandt, Alexander"https://www.zbmath.org/authors/?q=ai:brandt.alexander"Moir, Robert H. C."https://www.zbmath.org/authors/?q=ai:moir.robert-h-c"Moreno Maza, Marc"https://www.zbmath.org/authors/?q=ai:moreno-maza.marcSummary: We discuss algorithms for pseudo-division and division with remainder of multivariate polynomials with sparse representation. This work is motivated by the computations of normal forms and pseudo-remainders with respect to regular chains. We report on the implementation of those algorithms with the BPAS library.
For the entire collection see [Zbl 1396.68014].On unimodular matrices of difference operators.https://www.zbmath.org/1453.654422021-02-27T13:50:00+00:00"Abramov, S. A."https://www.zbmath.org/authors/?q=ai:abramov.sergei-a"Khmelnov, D. E."https://www.zbmath.org/authors/?q=ai:khmelnov.dennis-eSummary: We consider matrices \(L\in\text{Mat}_n(K[\sigma,\sigma^{-1}])\) of scalar difference operators, where \(K\) is a difference field of characteristic 0 with an automorphism \(\sigma\). We discuss approaches to compute the dimension of the space of those solutions of the system of equations \(L(y)=0\) that belong to an adequate extension of \(K\). On the base of one of those approaches, we propose a new algorithm for computing \(L^{-1}\in\text{Mat}_n(K[\sigma,\sigma^{-1}])\) whenever it exists. We investigate the worst-case complexity of the new algorithm, counting both arithmetic operations in \(K\) and shifts of elements of \(K\). This complexity turns out to be smaller than in the earlier proposed algorithms for inverting matrices of difference operators.
Some experiments with our implementation in Maple of the algorithm are reported.
For the entire collection see [Zbl 1396.68014].Some new integrals involving \(S\)-function and polynomials.https://www.zbmath.org/1453.330052021-02-27T13:50:00+00:00"Chand, Mehar"https://www.zbmath.org/authors/?q=ai:chand.meharSummary: In this paper, we aim to establish the new integrals involving \(S\)-function and Laguerre polynomials. On account of the most general nature of the functions involved herein, our main findings are capable of yielding a large number of new, interesting and useful integrals, expansion formula involving the \(S\)-function and the Laguerre polynomials as their special cases.A modified coordinate search method based on axes rotation.https://www.zbmath.org/1453.651322021-02-27T13:50:00+00:00"Chakraborty, Suvra Kanti"https://www.zbmath.org/authors/?q=ai:chakraborty.suvra-kanti"Panda, Geetanjali"https://www.zbmath.org/authors/?q=ai:panda.geetanjaliSummary: In this paper, a traditional coordinate search method is modified through rotation of axes and an expansion of square-stencil to capture the solution in a better and faster way. The scheme remains derivative free with global convergence property. The iterative process is explained for two-dimensional function in detail, which is followed by its extension to higher dimensions. Numerical illustrations and graphical representations for the sequential progress of the proposed scheme are provided. The comparison with the traditional coordinate search schemes through performance profiles are also provided to coin the advantages of the proposed scheme.
For the entire collection see [Zbl 1446.65004].Semi-implicit methods for the dynamics of elastic sheets.https://www.zbmath.org/1453.740812021-02-27T13:50:00+00:00"Alben, Silas"https://www.zbmath.org/authors/?q=ai:alben.silas"Gorodetsky, Alex A."https://www.zbmath.org/authors/?q=ai:gorodetsky.alex-a"Kim, Donghak"https://www.zbmath.org/authors/?q=ai:kim.donghak"Deegan, Robert D."https://www.zbmath.org/authors/?q=ai:deegan.robert-dSummary: Recent applications (e.g. active gels and self-assembly of elastic sheets) motivate the need to efficiently simulate the dynamics of thin elastic sheets. We present semi-implicit time stepping algorithms to improve the time step constraints that arise in explicit methods while avoiding much of the complexity of fully-implicit approaches. For a triangular lattice discretization with stretching and bending springs, our semi-implicit approach involves discrete Laplacian and biharmonic operators, and is stable for all time steps in the case of overdamped dynamics. For a more general finite-difference formulation that can allow for general elastic constants, we use the analogous approach on a square grid, and find that the largest stable time step is two to three orders of magnitude greater than for an explicit scheme. For a model problem with a radial traveling wave form of the reference metric, we find transitions from quasi-periodic to chaotic dynamics as the sheet thickness is reduced, wave amplitude is increased, and damping constant is reduced.PageRank computation with MAAOR and lumping methods.https://www.zbmath.org/1453.650772021-02-27T13:50:00+00:00"Mendes, I. R."https://www.zbmath.org/authors/?q=ai:mendes.i-r"Vasconcelos, P. B."https://www.zbmath.org/authors/?q=ai:vasconcelos.paulo-beleza|vasconcelos.pedro-bSummary: PageRank is a numerical method that Google uses to compute a page's importance, by assigning a score to every web page. PageRank is thus at the basis of Google's search engine success and can be mathematically explored either as an eigenvalue problem or as the solution of a homogeneous linear system. In both cases the Google matrix involved is large and sparse, so tuned algorithms must be developed to tackle it with the lowest computational cost and minimum memory requirements. One of such tunings is the Lumping method approach. Furthermore, the accuracy of the ranking vector needs not to be very precise, so inexpensive iterative methods are preferred. In this work the recent Matrix Analogue of the Accelerated Overrelaxation (MAAOR) iterative method is explored for the PageRank computation. Additionally Lumping methods have been applied to the eigenproblem formulation and we propose a novel approach combining the Lumping and MAAOR methods for the solution of the linear system. Numerical experiments illustrating the MAAOR method and the MAAOR method combined with Lumping techniques applied to PageRank computations are presented.Graph Merriman-Bence-Osher as a semidiscrete implicit Euler scheme for graph Allen-Cahn flow.https://www.zbmath.org/1453.652092021-02-27T13:50:00+00:00"Budd, Jeremy"https://www.zbmath.org/authors/?q=ai:budd.jeremy"Van Gennip, Yves"https://www.zbmath.org/authors/?q=ai:van-gennip.yvesThis article discusses a link between formulations of the Merriman-Bence-Osher scheme for diffusion generated motion and the Allen-Cahn gradient flow of the Ginzburg-Landau functional on finite graphs. Further results linking these flows with a graph formulation of mean curvature flow are discussed.
Reviewer: Marius Ghergu (Dublin)A conservation difference scheme of generalized Boussinesq equation.https://www.zbmath.org/1453.652212021-02-27T13:50:00+00:00"Jiang, Xiaoli"https://www.zbmath.org/authors/?q=ai:jiang.xiaoli"Wang, Xiaofeng"https://www.zbmath.org/authors/?q=ai:wang.xiaofeng.4Summary: We focus on the algorithm research of a class of six-order generalized Boussinesq equation. We use the finite difference method to discrete the Boussinesq equation. The discrete format with the law of energy conservation is deduced; stability and existence and good order of convergence properties are also derived. The efficiency of the proposed method is tested to numerical results that the convergence of space is of second-order and the conservation law of energy is verified very well for the energy difference.Investigation of stochastic nonlinear dynamics of ocean engineering systems through path integration.https://www.zbmath.org/1453.650242021-02-27T13:50:00+00:00"Varghese, Vini Anna"https://www.zbmath.org/authors/?q=ai:varghese.vini-anna"Saha, Nilanjan"https://www.zbmath.org/authors/?q=ai:saha.nilanjanSummary: Offshore structures operate in an uncertain environment under unsteady wave loads and also involve geometrical and material imperfections as well as parametric uncertainties. Additionally, there is sampling uncertainty and statistical uncertainty inherent in the response of these structures. These uncertainties have a significant impact on forecasting of the response unless the uncertainty is handled in a probabilistic manner. The present study tries to underline the importance by modelling the uncertainty through Weiner process by recasting the differential equations as stochastic differential ones. The stochastic differential equations are solved in an Ito framework using the Milstein scheme of solution. The study shows that there is a significant variation vis-à-vis the deterministic responses unless the modelling is done in a proper framework. Some of the sea states for numerical exposition are chosen which represent rogue waves. The importance of such framework is shown through three numerical illustrations: the response of tension leg platform with various parametric cases, a Mathieu-type instability problem and a three degree of freedom jacket structure. The efficacy of the Milstein method in handling a range of ocean nonlinear problems is brought out through these illustrations.Less is often more: applied inverse problems using \(hp\)-forward models.https://www.zbmath.org/1453.740802021-02-27T13:50:00+00:00"Smyl, Danny"https://www.zbmath.org/authors/?q=ai:smyl.danny"Liu, Dong"https://www.zbmath.org/authors/?q=ai:liu.dongSummary: To solve an applied inverse problem, a numerical forward model for the problem's physics is required. Commonly, the finite element method is employed with discretizations consisting of elements with variable size \(h\) and polynomial degree \(p\). Solutions to \(hp\)-forward models are known to converge exponentially by simultaneously decreasing \(h\) and increasing \(p\). On the other hand, applied inverse problems are often ill-posed and their minimization rate exhibits uncertainty. Presently, the behavior of applied inverse problems incorporating \(hp\) elements of differing \(p, h\), and geometry is not fully understood. Nonetheless, recent research suggests that employing increasingly higher-order \(hp\)-forward models (increasing mesh density and \(p)\) decreases reconstruction errors compared to inverse regimes using lower-order \(hp\)-forward models (coarser meshes and lower \(p)\). However, an affirmative or negative answer to following question has not been provided, ``Does the use of higher order \(hp\)-forward models in applied inverse problems always result in lower error reconstructions than approaches using lower order \(hp\)-forward models?'' In this article we aim to reduce the current knowledge gap and answer the open question by conducting extensive numerical investigations in the context of two contemporary applied inverse problems: elasticity imaging and hydraulic tomography -- nonlinear inverse problems with a PDE describing the underlying physics. Our results support a \textit{negative} answer to the question -- i.e. decreasing \(h\) (increasing mesh density), increasing \(p\), or simultaneously decreasing \(h\) and increasing \(p\) does not guarantee lower error reconstructions in applied inverse problems. Rather, there is complex balance between the accuracy of the \(hp\)-forward model, noise, prior knowledge (regularization), Jacobian accuracy, and ill-conditioning of the Jacobian matrix which ultimately contribute to reconstruction errors. As demonstrated herein, it is often more advantageous to use lower-order \(hp\)-forward models than higher-order \(hp\)-forward models in applied inverse problems. These realizations and other counterintuitive behavior of applied inverse problems using \(hp\)-forward models are described in detail herein.Numerical simulation of one-dimensional fractional nonsteady heat transfer model based on the second kind Chebyshev wavelet.https://www.zbmath.org/1453.653712021-02-27T13:50:00+00:00"Zhao, Fuqiang"https://www.zbmath.org/authors/?q=ai:zhao.fuqiang"Xie, Jiaquan"https://www.zbmath.org/authors/?q=ai:xie.jiaquan"Huang, Qingxue"https://www.zbmath.org/authors/?q=ai:huang.qingxueSummary: In the current study, a numerical technique for solving one-dimensional fractional nonsteady heat transfer model is presented. We construct the second kind Chebyshev wavelet and then derive the operational matrix of fractional-order integration. The operational matrix of fractional-order integration is utilized to reduce the original problem to a system of linear algebraic equations, and then the numerical solutions obtained by our method are compared with those obtained by CAS wavelet method. Lastly, illustrated examples are included to demonstrate the validity and applicability of the technique.Enhancement of ABCs efficiency at computer simulation of optical pulse interaction with inhomogeneous nonlinear medium.https://www.zbmath.org/1453.652352021-02-27T13:50:00+00:00"Trofimov, Vyacheslav A."https://www.zbmath.org/authors/?q=ai:trofimov.vyacheslav-a"Trykin, Evgeny M."https://www.zbmath.org/authors/?q=ai:trykin.evgeny-mSummary: We focus our attention on enhancing of computation performance at numerical simulation of a laser pulse interaction with inhomogeneous nonlinear medium (optical periodic structure or photonic crystal) which is described in the framework of 1D Schrödinger equation without using slowly varying envelope in spatial coordinate. It allows taking into account the optical pulse reflection from medium inhomogeneous. In practice, as a rule at computer simulation the domain size before photonic crystal is many times greater than the photonic crystal size. Decreasing this domain one can essentially increase a computation performance. This aim is achieved by re-statement of the problem: instead of initial distribution of complex amplitude along spatial coordinate before the photonic crystal we consider its specifying as a boundary condition (BC), which results in the pulse propagation in both directions of the propagation axis. Taking this into account and that a part of a laser pulse reflects from faces of photonic crystal layers, we use the artificial boundary conditions (ABCs). To avoid an influence of waves propagated in negative direction of spatial coordinate on the laser pulse interaction in photonic crystal (this influence appears due to imperfection of ABC). We introduce in consideration some number of artificial waves related with the equation under consideration. With this aim, firstly, we provide a computation of the laser pulse propagation in a linear medium and storage the complex amplitude values at chosen section of a coordinate, along which a laser light propagates. Then we use this complex amplitude values as a left boundary condition for 1D nonlinear Schrödinger equation in domain containing a nonlinear photonic crystal. To decrease amplitude of the wave reflected from artificial boundaries. We demonstrate also that instead of a complex amplitude values storage one can use the corresponding analytical expression for the linear Schrödinger equation solution, corresponding to a laser pulse propagation in a linear domain before the photonic crystal. We discuss also various ways for full transform of the optical pulse energy for reflected wave into artificial waves. This results essentially increasing of computation efficiency.Desingularization of bounded-rank matrix sets.https://www.zbmath.org/1453.650952021-02-27T13:50:00+00:00"Khrulkov, Valentin"https://www.zbmath.org/authors/?q=ai:khrulkov.valentin"Oseledets, Ivan"https://www.zbmath.org/authors/?q=ai:oseledets.ivan-vThe distance function from a real algebraic variety.https://www.zbmath.org/1453.650462021-02-27T13:50:00+00:00"Ottaviani, Giorgio"https://www.zbmath.org/authors/?q=ai:ottaviani.giorgio-maria"Sodomaco, Luca"https://www.zbmath.org/authors/?q=ai:sodomaco.lucaSummary: For any (real) algebraic variety \(X\) in a Euclidean space \(V\) endowed with a nondegenerate quadratic form \(q\), we introduce a polynomial \(\operatorname{EDpoly}_{X , u}( t^2)\) which, for any \(u \in V\), has among its roots the distance from \(u\) to \(X\). The degree of \(\operatorname{EDpoly}_{X , u}\) is the \textit{Euclidean Distance degree} of \(X\). We prove a duality property when \(X\) is a projective variety, namely \(\operatorname{EDpoly}_{X , u}( t^2) = \operatorname{EDpoly}_{X^\vee , u}(q(u) - t^2)\) where \(X^\vee\) is the dual variety of \(X\). When \(X\) is transversal to the isotropic quadric \(Q\), we prove that the ED polynomial of \(X\) is monic and the zero locus of its lower term is \(X \cup ( X^\vee \cap Q )^\vee \).Symbolic algorithm for generating the orthonormal Bargmann-Moshinsky basis for \(\mathrm{SU}(3)\) group.https://www.zbmath.org/1453.650812021-02-27T13:50:00+00:00"Deveikis, A."https://www.zbmath.org/authors/?q=ai:deveikis.algirdas"Gusev, A. A."https://www.zbmath.org/authors/?q=ai:gusev.alexander-a"Gerdt, V. P."https://www.zbmath.org/authors/?q=ai:gerdt.vladimir-p"Vinitsky, S. I."https://www.zbmath.org/authors/?q=ai:vinitsky.sergue-i"Góźdź, A."https://www.zbmath.org/authors/?q=ai:gozdz.andrzej"Pȩdrak, A."https://www.zbmath.org/authors/?q=ai:pedrak.aSummary: A symbolic algorithm which can be implemented in any computer algebra system for generating the Bargmann-Moshinsky (BM) basis with the highest weight vectors of \(\mathrm{SU}(3)\) irreducible representations is presented. The effective method resulting in analytical formula of overlap integrals in the case of the non-canonical BM basis [\textit{S. Alisauskas}, \textit{P. Raychev} and \textit{R. Roussev}, ``Analytical form of the orthonormal basis of the decomposition \(\mathrm{SU}(3)\supset \mathrm{O}(3)\supset \mathrm{O}(2)\) for some \((\lambda,\mu)\) multiplets'', J. Phys. G, Nucl. Phys. 7, 1213--1226 (1981; \url{doi:10.1088/0305-4616/7/9/013})] is used. A symbolic recursive algorithm for orthonormalisation of the obtained basis is developed. The effectiveness of the algorithms implemented in Mathematica 10.1 is investigated by calculation of the overlap integrals for up to \(\mu=5\) with \(\lambda>\mu\) and orthonormalization of the basis for up to \(\mu=4\) with \(\lambda>\mu\). The action of the zero component of the quadrupole operator onto the basis vectors with \(\mu=4\) is also obtained.
For the entire collection see [Zbl 1396.68014].A mixed-type Galerkin variational formulation and fast algorithms for variable-coefficient fractional diffusion equations.https://www.zbmath.org/1453.651952021-02-27T13:50:00+00:00"Li, Yongshan"https://www.zbmath.org/authors/?q=ai:li.yongshan"Chen, Huanzhen"https://www.zbmath.org/authors/?q=ai:chen.huanzhen"Wang, Hong"https://www.zbmath.org/authors/?q=ai:wang.hong.1Authors' abstract: We consider the variable-coefficient fractional diffusion equations with two-sided fractional derivative. By introducing an intermediate variable, we propose a mixed-type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over \(H^1_0(\Omega)\times H^{\frac{1-\beta}{2}}(\Omega)\). On the basis of the formulation, we develop a mixed-type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz-like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from \(O(N^2)\) to \(O(N)\) and the computing cost from \(O(N^3)\) to \(O(N\log N)\). Numerical experiments are included to verify our theoretical findings.
Reviewer: Li Changpin (Logan)Testing the multi-step single-stage method on stiff problems.https://www.zbmath.org/1453.651542021-02-27T13:50:00+00:00"Prusov, V. A."https://www.zbmath.org/authors/?q=ai:prusov.vitaliy-a"Doroshenko, A. Yu."https://www.zbmath.org/authors/?q=ai:doroshenko.anatoliy-yuSummary: A multi-step single-stage method is considered, which allows one to integrate stiff differential equations and systems of equations with high accuracy and low computational costs. The examples show that the proposed method is in solving stiff problems as good as the best available methods. The calculation results allow us to determine the absolute stability domains for the multi-step single-stage method, where it is possible to vary integration step within a wide range while maintaining the computational stability of the method.Adaptive space-time isogeometric analysis for parabolic evolution problems.https://www.zbmath.org/1453.653302021-02-27T13:50:00+00:00"Langer, Ulrich"https://www.zbmath.org/authors/?q=ai:langer.ulrich"Matculevich, Svetlana"https://www.zbmath.org/authors/?q=ai:matculevich.svetlana"Repin, Sergey"https://www.zbmath.org/authors/?q=ai:repin.sergey-iSummary: The paper proposes new locally stabilized space-time isogeometric analysis approximations to initial boundary value problems of the parabolic type. Previously, similar schemes (but weighted with a global mesh parameter) have been presented and studied by U. Langer, M. Neumüller, and S. Moore [\textit{U. Langer} et al., Comput. Methods Appl. Mech. Eng. 306, 342--363 (2016; Zbl 1436.76027)]. The current work devises a localized version of this scheme, which is suited for adaptive mesh refinement. We establish coercivity, boundedness, and consistency of the corresponding bilinear form. Using these fundamental properties together with standard approximation error estimates for B-splines and NURBS, we show that the space-time isogeometric analysis solutions generated by the new scheme satisfy asymptotically optimal a priori discretization error estimates. Error indicators used for mesh refinement are based on a posteriori error estimates of the functional type that have been introduced by \textit{S. I. Repin} and \textit{M. E. Frolov} [Zh. Vychisl. Mat. Mat. Fiz. 42, No. 12, 1774--1787 (2002; Zbl 1116.65324); translation in Comput. Math. Math. Phys. 42, No. 12, 1704--1716 (2002)], and later rigorously studied in the context of isogeometric analysis by U. Langer, S. Matculevich, and S. Repin [\textit{U. Langer} et al., Radon Ser. Comput. Appl. Math. 25, 141--183 (2019; Zbl 07224809)]. Numerical results discussed in the paper illustrate an improved convergence of global approximation errors and respective error majorants. They also confirm the local efficiency of the error indicators produced by the error majorants.
For the entire collection see [Zbl 1425.65008].A \(C^0 P_2\) time-stepping virtual element method for linear wave equations on polygonal meshes.https://www.zbmath.org/1453.653242021-02-27T13:50:00+00:00"Huang, Jianguo"https://www.zbmath.org/authors/?q=ai:huang.jianguo"Lin, Sen"https://www.zbmath.org/authors/?q=ai:lin.senThe authors propose and analyze the numerical method for the linear wave equation with two state variables. The proposed method uses the virtual element method for the spatial discretization in combination with the \(C^0P_2\) time-stepping scheme, i.e., a scheme that leads to a piecewise-quadratic solution with respect to time, which is continuous at time nodes. Using the energy method, the authors derive the optimal error estimates in the \(L^2\)-norm and the \(H^1\)-seminorm. The results of the numerical experiments are provided to illustrate the efficiency and applicability of the method.
Reviewer: Dana Černá (Liberec)A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation.https://www.zbmath.org/1453.653382021-02-27T13:50:00+00:00"Manzanero, Juan"https://www.zbmath.org/authors/?q=ai:manzanero.juan"Rubio, Gonzalo"https://www.zbmath.org/authors/?q=ai:rubio.gonzalo"Kopriva, David A."https://www.zbmath.org/authors/?q=ai:kopriva.david-a"Ferrer, Esteban"https://www.zbmath.org/authors/?q=ai:ferrer.esteban"Valero, Eusebio"https://www.zbmath.org/authors/?q=ai:valero.eusebioSummary: We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard equation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The latter permits us to show that the discrete free-energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface fluxes, and a first order IMplicit-EXplicit (IMEX) scheme to integrate in time. We provide a semi-discrete stability study, and a fully-discrete proof subject to a positivity condition on the solution. Lastly, we test the theoretical findings using numerical cases that include two and three-dimensional problems.Numerical simulation for 3D flow in flow channel of aeroengine turbine fan based on dimension splitting method.https://www.zbmath.org/1453.653272021-02-27T13:50:00+00:00"Ju, Guoliang"https://www.zbmath.org/authors/?q=ai:ju.guoliang"Chen, Can"https://www.zbmath.org/authors/?q=ai:chen.can|chen.can.1"Chen, Rongliang"https://www.zbmath.org/authors/?q=ai:chen.rongliang"Li, Jingzhi"https://www.zbmath.org/authors/?q=ai:li.jingzhi"Li, Kaitai"https://www.zbmath.org/authors/?q=ai:li.kaitai.1|li.kaitai"Zhang, Shaohui"https://www.zbmath.org/authors/?q=ai:zhang.shaohuiSummary: In this paper, we introduce a dimension splitting method for simulating the air flow state of the aeroengine turbine fan. Based on the geometric model of the fan blade, the dimension splitting method establishes a semi-geodesic coordinate system. Under such coordinate system, the Navier-Stokes equations are reformulated into the combination of membrane operator equations on two-dimensional manifolds and bending operator equations along the hub circle. Using Euler central difference scheme to approximate the third variable, the new form of Navier-Stokes equations is splitting into a set of two-dimensional sub-problems. Solving these sub-problems by alternate iteration, it follows an approximate solution to Navier-Stokes equations. Furthermore, we conduct a numerical experiment to show that the dimension splitting method has a good performance by comparing with the traditional methods. Finally, we give the simulation results of the pressure and flow state of the fan blade.Quadrature formulas of Gauss type for a sphere with nodes characterized by regular prism symmetry.https://www.zbmath.org/1453.650522021-02-27T13:50:00+00:00"Voloshchenko, A. M."https://www.zbmath.org/authors/?q=ai:voloshchenko.a-m"Russkov, A. A."https://www.zbmath.org/authors/?q=ai:russkov.a-aSummary: When the transport equation is solved by the discrete ordinate method, the problem arises of constructing quadrature formulas on a sphere characterized by the required accuracy and making it possible to use the quadrature nodes to approximate the transport equation in \(r\), \(\vartheta\), \(z\) geometry, in which quadrature nodes are simultaneously used to approximate the derivative with respect to the azimuth angle \(\varphi\) of the transport equation, that is, must be located in levels on the sphere with the same values of the polar angle \(\theta \). An algorithm is considered to construct quadrature formulas of the needed form that are characterized by regular prism (dihedron) symmetry and exact for all spherical polynomials of degree not exceeding some maximal value \(L\). This study is a development of the work of \textit{A. N. Kazakov} and \textit{V. I. Lebedev} [Proc. Steklov Inst. Math. 203, 89--99 (1995; Zbl 1126.41302); translation from Tr. Mat. Inst. Steklova 203, 100--112 (1994)]. The constructed family of quadratures, unlike that in the above work, does not contain nodes with \(\varphi = 0,\pi/2, \pi 3\pi/2\), at the poles \(\theta = \pm \pi/2\), and on the equator \(\theta = 0\) of the sphere. It is shown that this family ensures a significant computational gain when radiation transport problems are solved in three-dimensional geometry.Block basis factorization for scalable kernel evaluation.https://www.zbmath.org/1453.650912021-02-27T13:50:00+00:00"Wang, Ruoxi"https://www.zbmath.org/authors/?q=ai:wang.ruoxi"Li, Yingzhou"https://www.zbmath.org/authors/?q=ai:li.yingzhou"Mahoney, Michael W."https://www.zbmath.org/authors/?q=ai:mahoney.michael-w"Darve, Eric"https://www.zbmath.org/authors/?q=ai:darve.ericCheney-Sharma-type operators on a triangle with two or three curved edges.https://www.zbmath.org/1453.410022021-02-27T13:50:00+00:00"Baboş, A."https://www.zbmath.org/authors/?q=ai:babos.alinaIn this paper the author constructs operators of Cheney-Sharma-type with several interpolation properties on triangles with curved edges (two and three, respectively). He studies the properties of this type of interpolation as well as its degree of accuracy. Some examples are given.
Reviewer: Antonio López-Carmona (Granada)Faster randomized block Kaczmarz algorithms.https://www.zbmath.org/1453.650742021-02-27T13:50:00+00:00"Necoara, Ion"https://www.zbmath.org/authors/?q=ai:necoara.ionEfficiency improvement of discrete-ordinates method for interfacial phonon transport by Gauss-Legendre integral for frequency domain.https://www.zbmath.org/1453.654712021-02-27T13:50:00+00:00"Ran, Xin"https://www.zbmath.org/authors/?q=ai:ran.xin"Wang, Moran"https://www.zbmath.org/authors/?q=ai:wang.moranSummary: The thermal conduction through interfaces plays an important role for optimization and new designs in micro/nano energy system. However, the available knowledge and understanding on the physical picture of phonon transport at interfaces are still insufficient. The discrete-ordinates method (DOM) has been recently applied to solve the equation of phonon radiative transfer (EPRT) but it still suffers from expensive computational cost, especially with interfaces of different materials. In the present work, we introduce the Gauss-Legendre quadrature for frequency domain to improve the efficiency of discrete-ordinates method between heterogeneous materials over the uniform discretization scheme. The present scheme is verified for cross-plane interfacial phonon transport, with comparisons with both theoretical solution and kinetic-type Monte Carlo method. One to two orders of magnitude improvements on CPU time have been obtained numerically.Numerical modeling of seismic waves by discontinuous spectral element methods.https://www.zbmath.org/1453.860132021-02-27T13:50:00+00:00"Antonietti, Paola F."https://www.zbmath.org/authors/?q=ai:antonietti.paola-francesca"Ferroni, Alberto"https://www.zbmath.org/authors/?q=ai:ferroni.alberto"Mazzieri, Ilario"https://www.zbmath.org/authors/?q=ai:mazzieri.ilario"Paolucci, Roberto"https://www.zbmath.org/authors/?q=ai:paolucci.roberto"Quarteroni, Alfio"https://www.zbmath.org/authors/?q=ai:quarteroni.alfio-m"Smerzini, Chiara"https://www.zbmath.org/authors/?q=ai:smerzini.chiara"Stupazzini, Marco"https://www.zbmath.org/authors/?q=ai:stupazzini.marcoSummary: We present a comprehensive review of discontinuous Galerkin spectral element (DGSE) methods on hybrid hexahedral/tetrahedral grids for the numerical modeling of the ground motion induced by large earthquakes. DGSE methods combine the exibility of discontinuous Galerkin methods to patch together, through a domain decomposition paradigm, spectral element blocks where high-order polynomials are used for the space discretization. This approach allows local adaptivity on discretization parameters, thus, improving the quality of the solution without affecting the computational costs. The theoretical properties of the semidiscrete formulation are also revised, including well-posedness, stability and error estimates. A discussion on the dissipation, dispersion and stability properties of the fully-discrete (in space and time) formulation is also presented. Here, space discretization is obtained based on employing the leap-frog time marching scheme. The capabilities of the present approach are demonstrated through a set of computations of realistic earthquake scenarios obtained using the code \texttt{SPEED} (\url{http://speed.mox.polimi.it}), an open-source code specifically designed for the numerical modeling of large-scale seismic events jointly developed at Politecnico di Milano by The Laboratory for Modeling and Scientific Computing MOX and by the Department of Civil and Environmental Engineering.Introduction to high performance scientific computing.https://www.zbmath.org/1453.650012021-02-27T13:50:00+00:00"Chopp, D. L."https://www.zbmath.org/authors/?q=ai:chopp.david-lThe book gives an introduction into tools and programming languages which one needs for performing high-performance scientific computing. In Chapter 1, some basic concepts, e.g., development environments, editors, compilers, and makefiles are shortly described. Part I of the book is devoted to an introduction of the programming language C. The structure of a C program, data types and structures, possibilities for input and output, flow control statements, functions, and the usage of libraries are in details explained and demonstrated. The topic of Part II is parallel programming for shared memory machines using OpenMP. Subdividing of loops is described, and OpenMP libraries are presented. In Part III the parallel programming on distributed memory machines using MPI is discussed. Especially, routines for the communication between the processes are explained in detail. Part IV is devoted to the GPU programming and to CUDA which is an extension to the programming language C. In Part V the GPU programming using OpenCL is described. In Part VI of the book some applications (stochastic differential equations, finite difference methods, iterative solution of elliptic equations and pseudospectral methods) are presented which can be solved by using the different approaches discussed in the book. Each chapter ends with some exercises.
Reviewer: Michael Jung (Dresden)Balanced truncation model order reduction in limited time intervals for large systems.https://www.zbmath.org/1453.650932021-02-27T13:50:00+00:00"Kürschner, Patrick"https://www.zbmath.org/authors/?q=ai:kurschner.patrickSummary: In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on the efficient numerical realization of this model reduction approach in case of large system dimensions. We discuss numerical methods to deal with the resulting matrix exponential functions and Lyapunov equations which are solved for low-rank approximations. Our main tool for this purpose are rational Krylov subspace methods. We also discuss the eigenvalue decay and numerical rank of the solutions of the Lyapunov equations. These results, and also numerical experiments, will show that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time-limited balanced truncation can be smaller compared to standard balanced truncation. In numerical experiments we test the approaches for computing low-rank factors of the involved Lyapunov solutions and illustrate that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region.On robust width property for Lasso and Dantzig selector.https://www.zbmath.org/1453.651392021-02-27T13:50:00+00:00"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui.8|zhang.hui.11|zhang.hui.2|zhang.hui.4|zhang.hui.3|zhang.hui.7|zhang.hui.9|zhang.hui.5|zhang.hui.10|zhang.hui.6|zhang.hui.1|zhang.huiSummary: Recently, Cahill and Mixon completely characterized sensing operators in many compressed sensing instances with a robust width property, which allows uniformly stable and robust reconstruction via certain convex optimization. However, their current theory does not cover the Lasso and Dantzig selector models, both of which are popular alternatives in statistics and optimization community. In this short note, we show that the robust width property can be perfectly applied to these two types of models as well. Our main results affirmatively answer the question left by Cahill and Mixon.Multiscale gentlest ascent dynamics for saddle point in effective dynamics of slow-fast system.https://www.zbmath.org/1453.601292021-02-27T13:50:00+00:00"Gu, Shuting"https://www.zbmath.org/authors/?q=ai:gu.shuting"Zhou, Xiang"https://www.zbmath.org/authors/?q=ai:zhou.xiang|zhou.xiang.1Summary: Here, we present a multiscale method to calculate the saddle point associated with the effective dynamics arising from a stochastic system which couples slow deterministic drift and fast stochastic dynamics. This problem is motivated by the transition states on free energy surfaces in chemical physics. Our method is based on the gentlest ascent dynamics which couples the position variable and the direction variable and has the local convergence to saddle points. The dynamics of the direction vector is derived in terms of the covariance function with respective to the equilibrium distribution of the fast stochastic process. We apply multiscale numerical methods to efficiently solve the obtained multiscale gentlest ascent dynamics, and discuss the acceleration techniques based on an adaptive idea. The examples of stochastic ordinary and partial differential equations are presented.Optimal strong approximation of the one-dimensional squared Bessel process.https://www.zbmath.org/1453.650182021-02-27T13:50:00+00:00"Hefter, Mario"https://www.zbmath.org/authors/?q=ai:hefter.mario"Herzwurm, André"https://www.zbmath.org/authors/?q=ai:herzwurm.andreSummary: We consider the one-dimensional squared Bessel process given by the stochastic differential equation (SDE)
\[
\mathrm dX_t = 1\mathrm dt + 2 \sqrt{X_t} \mathrm{d}W_t, \qquad X_0 = x_0, \qquad t \in [0,1],
\]
and study strong (pathwise) approximation of the solution \(X\) at the final time point \(t=1\). This SDE is a particular instance of a Cox-Ingersoll-Ross (CIR) process where the boundary point zero is accessible. We consider numerical methods that have access to values of the driving Brownian motion \(W\) at a finite number of time points. We show that the polynomial convergence rate of the \(n\)-th minimal errors for the class of adaptive algorithms as well as for the class of algorithms that rely on equidistant grids are equal to infinity and \(1/2\), respectively. This shows that adaption results in a tremendously improved convergence rate. As a by-product, we obtain that the parameters appearing in the CIR process affect the convergence rate of strong approximation.A parameter estimation method for stiff ordinary differential equations using particle swarm optimisation.https://www.zbmath.org/1453.651502021-02-27T13:50:00+00:00"Arloff, William"https://www.zbmath.org/authors/?q=ai:arloff.william"Schmitt, Karl R. B."https://www.zbmath.org/authors/?q=ai:schmitt.karl-r-b"Venstrom, Luke J."https://www.zbmath.org/authors/?q=ai:venstrom.luke-jSummary: We propose a two-step method for fitting stiff ordinary differential equation (ODE) models to experimental data. The first step avoids integrating stiff ODEs during the unbounded search for initial estimates of model parameters. To avoid integration, a polynomial approximation of experimental data is generated, differentiated and compared directly to the ODE model, obtaining crude but physically plausible estimates for model parameters. Particle swarm optimisation (PSO) is used for the parameter search to overlook combinations of model parameters leading to undefined solutions of the stiff ODE. After initial estimates are determined, the second step numerically solves the ODE. This refines model parameter values through a bounded search. We demonstrate this method by fitting the model parameters (activation energies and pre-exponential factors) of the Arrhenius-based temperature-dependent kinetic coefficients in the shrinking core solid-state chemical kinetics model for the reduction of Cobalt (II, III) Oxide (Co\(_3\)O\(_4\)) particles to Cobalt (II) Oxide (CoO).Optimized quasiconformal parameterization with user-defined area distortions.https://www.zbmath.org/1453.650442021-02-27T13:50:00+00:00"Lam, Ka Chun"https://www.zbmath.org/authors/?q=ai:lam.ka-chun"Lui, Lok Ming"https://www.zbmath.org/authors/?q=ai:lui.lok-mingSummary: Parameterization, a process of mapping a complicated domain onto a simple canonical domain, is crucial in different areas such as computer graphics, medical imaging and scientific computing. Conformal parameterization has been widely used since it preserves the local geometry well. However, a major drawback is the area distortion introduced by the conformal parameterization, causing inconvenience in many applications such as texture mapping in computer graphics or visualization in medical imaging. This work proposes a remedy to construct a parameterization that balances between conformality and area distortions. We present a variational algorithm to compute the optimized quasiconformal parameterization with controllable area distortions. The distribution of the area distortion can be prescribed by users according to the application. The main strategy is to minimize a combined energy functional consisting of an area mismatching term and a regularization term involving the Beltrami coefficient of the map. The Beltrami coefficient controls the conformality of the parameterization. Landmark constraints can be incorporated into the model to obtain landmark-aligned parameterization. Experiments have been carried out on both synthetic and real data. Results demonstrate the efficacy of the proposed algorithm to compute the optimized parameterization with controllable area distortion while preserving the local geometry as well as possible.A low-rank approximated multiscale method for PDEs with random coefficients.https://www.zbmath.org/1453.654352021-02-27T13:50:00+00:00"Ou, Na"https://www.zbmath.org/authors/?q=ai:ou.na"Lin, Guang"https://www.zbmath.org/authors/?q=ai:lin.guang"Jiang, Lijian"https://www.zbmath.org/authors/?q=ai:jiang.lijianEnsemble Kalman filter for multiscale inverse problems.https://www.zbmath.org/1453.653932021-02-27T13:50:00+00:00"Abdulle, Assyr"https://www.zbmath.org/authors/?q=ai:abdulle.assyr"Garegnani, Giacomo"https://www.zbmath.org/authors/?q=ai:garegnani.giacomo"Zanoni, Andrea"https://www.zbmath.org/authors/?q=ai:zanoni.andreaLegendre wavelets direct method for the numerical solution of time-fractional order telegraph equations.https://www.zbmath.org/1453.653692021-02-27T13:50:00+00:00"Xu, Xiaoyong"https://www.zbmath.org/authors/?q=ai:xu.xiaoyong"Xu, Da"https://www.zbmath.org/authors/?q=ai:xu.daSummary: In this paper, a Legendre wavelet collocation method for solving a class of time-fractional order telegraph equation defined by Caputo sense is discussed. Fractional integral formula of a single Legendre wavelet in the Riemann-Liouville sense is derived by means of shifted Legendre polynomials. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence analysis and error analysis of the proposed method are investigated. Several examples are presented to show the applicability and accuracy of the proposed method.Misfit function for full waveform inversion based on the Wasserstein metric with dynamic formulation.https://www.zbmath.org/1453.860312021-02-27T13:50:00+00:00"Yong, Peng"https://www.zbmath.org/authors/?q=ai:yong.peng"Liao, Wenyuan"https://www.zbmath.org/authors/?q=ai:liao.wenyuan"Huang, Jianping"https://www.zbmath.org/authors/?q=ai:huang.jianping"Li, Zhenchun"https://www.zbmath.org/authors/?q=ai:li.zhenchun"Lin, Yaoting"https://www.zbmath.org/authors/?q=ai:lin.yaotingSummary: Conventional full waveform inversion (FWI) using least square distance \((L_2\) norm) between the observed and predicted seismograms suffers from local minima. Recently, the Wasserstein metric \((W_1\) metric) has been introduced to FWI to compute the misfit between two seismograms. Instead of comparisons bin by bin, the \(W_1\) metric allows to compare signal intensities across different coordinates. This measure has great potential to account for time and space shifts of events within seismograms. However, there are two main challenges in application of the \(W_1\) metric to FWI. The first one is that the compared signals need to satisfy nonnegativity and mass conservation assumptions. The second one is that the computation of \(W_1\) metric between two seismograms is a computationally expensive problem. In this paper, a strategy is used to satisfy the two assumptions via decomposition and recombination of original seismic data. In addition, the computation of the \(W_1\) metric based on dynamic formulation is formulated as a convex optimization problem. A primal-dual hybrid gradient method with linesearch has been developed to solve this large-scale optimization problem on GPU device. The advantages of the new method are that it is easy to implement and has high computational efficiency. Compared to the \(L_2\) norm based FWI, the computation time of the proposed method will approximately increase by 11\% in our case studies. A 1D time-shift signals case study has indicated that the \(W_1\) metric is more effective in capturing time shift and makes the misfit function more convex. Two applications to synthetic data using transmissive and reflective recording geometries have demonstrated the effectiveness of the \(W_1\) metric in mitigating cycle-skipping issues. We have also applied the proposed method to SEG 2014 benchmark data, which has further demonstrated that the \(W_1\) metric can mitigate local minima and provide reliable velocity estimations without using low frequency information in the recorded data.A boundary element method for homogenization of periodic structures.https://www.zbmath.org/1453.654242021-02-27T13:50:00+00:00"Lukáš, Dalibor"https://www.zbmath.org/authors/?q=ai:lukas.dalibor"Of, Günther"https://www.zbmath.org/authors/?q=ai:of.gunther"Zapletal, Jan"https://www.zbmath.org/authors/?q=ai:zapletal.jan"Bouchala, Jiří"https://www.zbmath.org/authors/?q=ai:bouchala.jiriThe scalar elliptic boundary value problem
\[
-\mathrm{div}\left(a_\epsilon(x)\nabla u_\epsilon(x)\right)=f(x)\,,\quad x\in\Omega,
\]
with Lipschitz domain \(\Omega\subset \mathbb{R}^d\), \(d=2,3\), is considered with a linear boundary condition. Here, \(f\) is the source term, \(\epsilon>0\) is a geometrical period, and \(a_\epsilon\) is a periodic material function. The problem is reformulated as a homogenized problem which needs to be solved within a periodic cell. The authors solve this problem numerically with the boundary integral equations using Steklov-Poincaré operators which is then discretized only on the boundary of the periodic cell and the interface between the materials within the cell. Next, the existence is proven and further that the method converges super-linearly with respect to the mesh size. Finally, numerical results in both two and three dimensions are provided supporting their theoretical results.
Reviewer: Andreas Kleefeld (Jülich)Special issue: Latest computational methods on fractional dynamic systems ``VSI fractional dynamic system''.https://www.zbmath.org/1453.000292021-02-27T13:50:00+00:00"Debbouche, Amar (ed.)"https://www.zbmath.org/authors/?q=ai:debbouche.amar"Fečkan, Michal (ed.)"https://www.zbmath.org/authors/?q=ai:feckan.michal"Hernández, Eduardo (ed.)"https://www.zbmath.org/authors/?q=ai:hernandez.eduardo-mFrom the text: This special issue, resulting after an open call for papers that considered substantially extended versions of papers presented at the conference ICMMAS'19 as well as external submissions.Solving systems of nonlinear equations via conjugate direction flower pollination algorithm.https://www.zbmath.org/1453.651112021-02-27T13:50:00+00:00"Rushdy, Ehab"https://www.zbmath.org/authors/?q=ai:rushdy.ehab"Abdel-Baset, Mohamed"https://www.zbmath.org/authors/?q=ai:abdelbaset.mohamed"Hezam, Ibrahim M."https://www.zbmath.org/authors/?q=ai:hezam.ibrahim-mSummary: In this paper, we propose a new hybrid algorithm for solving system of nonlinear equations. The aim of hybridisation is to utilise the feature of flower pollination algorithm (FPA) and conjugate direction method (CD). Conjugate direction flower pollination algorithm (CDFPA) combines the advantages of CD and FPA. The problem of solving nonlinear equations is equivalently changed to the problem of function optimisation and then a solution is obtained by CDFPA. The results show the proposed algorithm has high convergence speed and accuracy for solving nonlinear equations.A note on preconditioner for the Ohta-Kawasaki equation.https://www.zbmath.org/1453.650662021-02-27T13:50:00+00:00"Li, Rui-Xia"https://www.zbmath.org/authors/?q=ai:li.ruixia"Liang, Zhao-Zheng"https://www.zbmath.org/authors/?q=ai:liang.zhao-zheng"Zhang, Guo-Feng"https://www.zbmath.org/authors/?q=ai:zhang.guofeng"Liao, Li-Dan"https://www.zbmath.org/authors/?q=ai:liao.lidan"Zhang, Lei"https://www.zbmath.org/authors/?q=ai:zhang.lei.24|zhang.lei.25Summary: In this paper, an efficient preconditioner is proposed for the discrete Ohta-Kawasaki system. We show that the eigenvalues of the new preconditioned matrix are the same as that of the recently published block triangular preconditioner in [\textit{P. E. Farrell} and \textit{J. W. Pearson}, SIAM J. Matrix Anal. Appl. 38, No. 1, 217--225 (2017; Zbl 1365.65221)]. Numerical experiments are presented to illustrate the effectiveness and feasibility of the new preconditioner to accelerate GMRES method.A finite element method for nematic liquid crystals with variable degree of orientation.https://www.zbmath.org/1453.820902021-02-27T13:50:00+00:00"Nochetto, Ricardo H."https://www.zbmath.org/authors/?q=ai:nochetto.ricardo-h"Walker, Shawn W."https://www.zbmath.org/authors/?q=ai:walker.shawn-w"Zhang, Wujun"https://www.zbmath.org/authors/?q=ai:zhang.wujunA local-global multiscale mortar mixed finite element method for multiphase transport in heterogeneous media.https://www.zbmath.org/1453.760692021-02-27T13:50:00+00:00"Fu, Shubin"https://www.zbmath.org/authors/?q=ai:fu.shubin"Chung, Eric T."https://www.zbmath.org/authors/?q=ai:chung.eric-tSummary: In this paper, we propose a local-global multiscale mortar mixed finite element method (MMMFEM) for multiphase transport in heterogeneous media. It is known that, in the efficient numerical simulations of this problem, one important step is a fast solution of the pressure equation, which is required to be solved in each time step. Thus, some types of efficient numerical methods, such as multiscale methods, are crucial for this problem. To present our main concepts, we take the two-phase flow system as an example. In our proposed method, the pressure equation is solved via the multiscale mortar mixed finite element method. Using this approach, a mass conservative velocity field can be obtained. Next, we use an explicit finite volume method to solve the saturation equation. The key ingredient of our proposed method is the choice of mortar space for the MMMFEM. We will use both polynomials and multiscale basis functions to form the coarse mortar space. The multiscale basis functions used are the restriction of the global pressure field obtained at the previous time step on the coarse interface. To initialize the simulations, we solve the pressure equation on the fine grid. We will present several numerical experiments on some benchmark 2D and 3D heterogeneous models to show the excellent performance of our method.The effects of unequal diffusion coefficients on spatiotemporal pattern formation in prey harvested reaction-diffusion systems.https://www.zbmath.org/1453.652162021-02-27T13:50:00+00:00"Guin, Lakshmi Narayan"https://www.zbmath.org/authors/?q=ai:guin.lakshmi-narayanSummary: In this investigation, we explore the spatiotemporal dynamics of reaction-diffusion predator-prey systems with Holling type II functional response. For partial differential equation, we consider the diffusion-driven instability of the coexistence equilibrium solution through spatiotemporal patterns. We find the conditions for Turing bifurcation of the system in a two-dimensional spatial domain by making use of the linear stability analysis and the bifurcation analysis. By choosing the ecological system parameter as the bifurcation parameter, we show that the system experiences a sequence of spatiotemporal patterns. The results of numerical simulations unveil that there are various spatial patterns including typical Turing patterns such as hot spots, spots-stripes mixture and stripes pattern through Turing instability. Our results show that the ecological system parameter plays a vital function in the proposed reaction-diffusion predator-prey models. Numerical design has been finally carried out through graphical representations of those outcomes towards the end in order to recognize the spatiotemporal behaviour of the system under study. All the outcomes are predictable to be of use in the study of the dynamic complexity of flora and fauna.
For the entire collection see [Zbl 1446.65004].A flux-conservative finite difference scheme for anisotropic bioelectric problems.https://www.zbmath.org/1453.921532021-02-27T13:50:00+00:00"Bourantas, George C."https://www.zbmath.org/authors/?q=ai:bourantas.george-c"Zwick, Benjamin F."https://www.zbmath.org/authors/?q=ai:zwick.benjamin-f"Warfield, Simon K."https://www.zbmath.org/authors/?q=ai:warfield.simon-k"Hyde, Damon E."https://www.zbmath.org/authors/?q=ai:hyde.damon-e"Wittek, Adam"https://www.zbmath.org/authors/?q=ai:wittek.adam"Miller, Karol"https://www.zbmath.org/authors/?q=ai:miller.karolSummary: We present a flux-conservative finite difference (FCFD) scheme for solving inhomogeneous anisotropic bioelectric problems. The method applies directly on the raw medical image data without the need for sophisticated image analysis algorithms to define interfaces between materials with different electrical conductivities. We demonstrate the accuracy of the method by comparison with analytical solution. Results for a patient-specific head model highlight the applicability of the method.
For the entire collection see [Zbl 1453.92005].Preface: Numerical solution of differential and differential-algebraic equations. Selected papers from NUMDIFF-15.https://www.zbmath.org/1453.000252021-02-27T13:50:00+00:00"Arnold, Martin (ed.)"https://www.zbmath.org/authors/?q=ai:arnold.martin"Podhaisky, Helmut (ed.)"https://www.zbmath.org/authors/?q=ai:podhaisky.helmut"Weiner, Rüdiger (ed.)"https://www.zbmath.org/authors/?q=ai:weiner.rudiger"Celledoni, Elena (ed.)"https://www.zbmath.org/authors/?q=ai:celledoni.elena"Frank, Jason (ed.)"https://www.zbmath.org/authors/?q=ai:frank.jason"Lang, Jens (ed.)"https://www.zbmath.org/authors/?q=ai:lang.jensFrom the text: This special issue contains a selection of papers presented at the 15th Seminar NUMDIFF on the Numerical Solution of Differential and Differential-Algebraic Equations, held at the Martin Luther University of Halle-Wittenberg, Halle, Germany, during 3--7 September 2018.Book review of: P. G. Constantine, Active subspaces. Emerging ideas for dimension reduction in parameter studies.https://www.zbmath.org/1453.000092021-02-27T13:50:00+00:00"Rozza, Gianluigi"https://www.zbmath.org/authors/?q=ai:rozza.gianluigiReview of [Zbl 1431.65001].Topological index analysis applied to coupled flow networks.https://www.zbmath.org/1453.652012021-02-27T13:50:00+00:00"Baum, Ann-Kristin"https://www.zbmath.org/authors/?q=ai:baum.ann-kristin"Kolmbauer, Michael"https://www.zbmath.org/authors/?q=ai:kolmbauer.michael"Offner, Günter"https://www.zbmath.org/authors/?q=ai:offner.gunterSummary: This work is devoted to the analysis of multi-physics dynamical systems stemming from automated modeling processes in system simulation software. The multi-physical model consists of (simple connected) networks of different or the same physical type (liquid flow, electric, gas flow, heat flow) which are connected via interfaces or coupling conditions. Since the individual networks result in differential algebraic equations (DAEs), the combination of them gives rise to a system of DAEs. While for the individual networks existence and uniqueness results, including the formulation of index reduced systems, is available through the techniques of \textit{modified nodal analysis} or \textit{topological based index analysis}, topological results for coupled system are not available so far. We present an approach for the application of topological based index analysis for coupled systems of the same physical type and give the outline of this approach for coupled liquid flow networks. Exploring the network structure via graph theoretical approaches allows to develop topological criteria for the existence of solutions of the coupled systems. The conditions imposed on the coupled network are illustrated via various examples. Those results can be interpreted as a natural extensions of the topological existence and index criteria provided by the topological analysis for single connected circuits.
For the entire collection see [Zbl 1419.65001].Exponential time differencing for mimetic multilayer Ocean models.https://www.zbmath.org/1453.860192021-02-27T13:50:00+00:00"Pieper, Konstantin"https://www.zbmath.org/authors/?q=ai:pieper.konstantin"Sockwell, K. Chad"https://www.zbmath.org/authors/?q=ai:sockwell.k-chad"Gunzburger, Max"https://www.zbmath.org/authors/?q=ai:gunzburger.max-dSummary: A framework for exponential time discretization of the multilayer rotating shallow water equations is developed in combination with a mimetic discretization in space. The method is based on a combination of existing exponential time differencing (ETD) methods and a careful choice of approximate Jacobians. The discrete Hamiltonian structure and conservation properties of the model are taken into account, in order to ensure stability of the method for large time steps and simulation horizons. In the case of many layers, further efficiency can be gained by a layer reduction which is based on the vertical structure of fast and slow modes. Numerical experiments on the example of a mid-latitude regional ocean model confirm long term stability for time steps increased by an order of magnitude over the explicit CFL, while maintaining accuracy for key statistical quantities.Adaptive spectral solution method for the Landau and Lenard-Balescu equations.https://www.zbmath.org/1453.653632021-02-27T13:50:00+00:00"Scullard, Christian R."https://www.zbmath.org/authors/?q=ai:scullard.christian-r"Hickok, Abigail"https://www.zbmath.org/authors/?q=ai:hickok.abigail"Sotiris, Justyna O."https://www.zbmath.org/authors/?q=ai:sotiris.justyna-o"Tzolova, Bilyana M."https://www.zbmath.org/authors/?q=ai:tzolova.bilyana-m"Van Heyningen, R. Loek"https://www.zbmath.org/authors/?q=ai:van-heyningen.r-loek"Graziani, Frank R."https://www.zbmath.org/authors/?q=ai:graziani.frank-rSummary: We present an adaptive spectral method for solving the Landau/Fokker-Planck equation for electron-ion systems. The heart of the algorithm is an expansion in Laguerre polynomials, which has several advantages, including automatic conservation of both energy and particles without the need for any special discretization or time-stepping schemes. One drawback of such an expansion is the \(O(N^3)\) memory requirement, where \(N\) is the number of polynomials used. This can impose an inconvenient limit in cases of practical interest, such as when two particle species have widely separated temperatures. The algorithm we describe here addresses this problem by periodically re-projecting the solution onto a judicious choice of new basis functions that are still Laguerre polynomials but have arguments adapted to the current physical conditions. This results in a reduction in the number of polynomials needed, at the expense of increased solution time. Because the equations are solved with little difficulty, this added time is not of much concern compared to the savings in memory. To demonstrate the algorithm, we solve several relaxation problems that could not be computed with the spectral method without re-projection. Another major advantage of this method is that it can be used for collision operators more complicated than that of the Landau equation, and we demonstrate this here by using it to solve the non-degenerate quantum Lenard-Balescu (QLB) equation for a hydrogen plasma. We conclude with some comparisons of temperature relaxation problems solved with the latter equation and the Landau equation with a Coulomb logarithm inspired by the properties of the QLB operator. We find that with this choice of Coulomb logarithm, there is little difference between using the two equations for these particular systems.Semi-discrete finite difference schemes for the nonlinear Cauchy problems of the normal form.https://www.zbmath.org/1453.651172021-02-27T13:50:00+00:00"Higashimori, Nobuyuki"https://www.zbmath.org/authors/?q=ai:higashimori.nobuyuki"Fujiwara, Hiroshi"https://www.zbmath.org/authors/?q=ai:fujiwara.hiroshiSummary: We consider the Cauchy problems of nonlinear partial differential equations of the normal form in the class of analytic functions. We apply semi-discrete finite difference approximation which discretizes the problems only with respect to the time variable, and we give a proof for its convergence. The result implies that there are cases of convergence of finite difference schemes applied to ill-posed Cauchy problems.Modified variational iteration algorithm-II: convergence and applications to diffusion models.https://www.zbmath.org/1453.653772021-02-27T13:50:00+00:00"Ahmad, Hijaz"https://www.zbmath.org/authors/?q=ai:ahmad.hijaz"Khan, Tufail A."https://www.zbmath.org/authors/?q=ai:khan.tufail-a"Stanimirović, Predrag S."https://www.zbmath.org/authors/?q=ai:stanimirovic.predrag-s"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yuming"Ahmad, Imtiaz"https://www.zbmath.org/authors/?q=ai:ahmad.imtiazSummary: Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.High-order gradients with the shifted boundary method: an embedded enriched mixed formulation for elliptic PDEs.https://www.zbmath.org/1453.760822021-02-27T13:50:00+00:00"Nouveau, L."https://www.zbmath.org/authors/?q=ai:nouveau.leo"Ricchiuto, M."https://www.zbmath.org/authors/?q=ai:ricchiuto.mario"Scovazzi, G."https://www.zbmath.org/authors/?q=ai:scovazzi.guglielmoSummary: We propose an extension of the embedded boundary method known as ``shifted boundary method'' to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded.Multi space reduced basis preconditioners for large-scale parametrized PDEs.https://www.zbmath.org/1453.654162021-02-27T13:50:00+00:00"Santo, Niccolò Dal"https://www.zbmath.org/authors/?q=ai:dal-santo.niccolo"Deparis, Simone"https://www.zbmath.org/authors/?q=ai:deparis.simone"Manzoni, Andrea"https://www.zbmath.org/authors/?q=ai:manzoni.andrea"Quarteroni, Alfio"https://www.zbmath.org/authors/?q=ai:quarteroni.alfio-mParallel hybrid sparse linear system solvers.https://www.zbmath.org/1453.650872021-02-27T13:50:00+00:00"Manguoğlu, Murat"https://www.zbmath.org/authors/?q=ai:manguoglu.murat"Polizzi, Eric"https://www.zbmath.org/authors/?q=ai:polizzi.eric"Sameh, Ahmed H."https://www.zbmath.org/authors/?q=ai:sameh.ahmed-hSummary: In this chapter, we present the SPIKE family of algorithms for solving banded linear systems and its multithreaded implementation as well as direct-iterative hybrid variants for solving general sparse linear system of equations.
For the entire collection see [Zbl 1446.65003].New finite volume weighted essentially nonoscillatory schemes on triangular meshes.https://www.zbmath.org/1453.652652021-02-27T13:50:00+00:00"Zhu, Jun"https://www.zbmath.org/authors/?q=ai:zhu.jun"Qiu, Jianxian"https://www.zbmath.org/authors/?q=ai:qiu.jianxianAccelerating the solution of linear systems by iterative refinement in three precisions.https://www.zbmath.org/1453.650672021-02-27T13:50:00+00:00"Carson, Erin"https://www.zbmath.org/authors/?q=ai:carson.erin"Higham, Nicholas J."https://www.zbmath.org/authors/?q=ai:higham.nicholas-jImplicit-explicit integral deferred correction methods for stiff problems.https://www.zbmath.org/1453.651532021-02-27T13:50:00+00:00"Boscarino, Sebastiano"https://www.zbmath.org/authors/?q=ai:boscarino.sebastiano"Qiu, Jing-Mei"https://www.zbmath.org/authors/?q=ai:qiu.jingmei"Russo, Giovanni"https://www.zbmath.org/authors/?q=ai:russo.giovanni.2|russo.giovanniStudy of instability of the Fourier split-step method for the massive Gross-Neveu model.https://www.zbmath.org/1453.652712021-02-27T13:50:00+00:00"Lakoba, T. I."https://www.zbmath.org/authors/?q=ai:lakoba.taras-iSummary: Stability properties of the well-known Fourier split-step method used to simulate a soliton and similar solutions of the nonlinear Dirac equations, known as the Gross-Neveu model, are studied numerically and analytically. Three distinct types of numerical instability that can occur in this case, are revealed and explained. While one of these types can be viewed as being related to the numerical instability occurring in simulations of the nonlinear Schrödinger equation, the other two types have not been studied or identified before, to the best of our knowledge. These two types of instability are \textit{unconditional}, i.e. occur for arbitrarily small values of the time step. They also persist in the continuum limit, i.e. for arbitrarily fine spatial discretization. Moreover, one of them persists in the limit of an infinitely large computational domain. It is further demonstrated that similar instabilities also occur for other numerical methods applied to the Gross-Neveu soliton, as well as to certain solitons of another relativistic field theory model, the massive Thirring.Low rank approximation of a sparse matrix based on LU factorization with column and row tournament pivoting.https://www.zbmath.org/1453.650902021-02-27T13:50:00+00:00"Grigori, Laura"https://www.zbmath.org/authors/?q=ai:grigori.laura"Cayrols, Sebastien"https://www.zbmath.org/authors/?q=ai:cayrols.sebastien"Demmel, James W."https://www.zbmath.org/authors/?q=ai:demmel.james-weldonA weakly compressible SPH method for violent multi-phase flows with high density ratio.https://www.zbmath.org/1453.761702021-02-27T13:50:00+00:00"Rezavand, Massoud"https://www.zbmath.org/authors/?q=ai:rezavand.massoud"Zhang, Chi"https://www.zbmath.org/authors/?q=ai:zhang.chi"Hu, Xiangyu"https://www.zbmath.org/authors/?q=ai:hu.xiangyuSummary: The weakly compressible SPH (WCSPH) method is known suffering from low computational efficiency, or unnatural voids and unrealistic phase separation when it is applied to simulate highly violent multi-phase flows with high density ratio, such as that between water and air. In this paper, to remedy these issues, we propose a multi-phase WCSPH method based on a low-dissipation Riemann solver and the transport-velocity formulation. The two-phase Riemann problem is first constructed to handle the pairwise interaction between fluid particles, then modified for the fluid-wall interaction to impose the solid wall boundary condition. Since the method uses the same artificial speed of sound for both heavy and light phases, the computational efficiency increases greatly. Furthermore, due to the transport-velocity formulation employed for the light phase and application of the two-phase Riemann problem, the unnatural voids and unrealistic phase separation are effectively eliminated. The method is validated with several 2- and 3D cases involving violent water-air flows, and demonstrates good robustness, improved or comparable accuracy, respectively, comparing to previous methods with the same choice of sound speed or those with much less computational efficiency.A comparison between alternating segment Crank-Nicolson and explicit-implicit schemes for the dispersive equation.https://www.zbmath.org/1453.652172021-02-27T13:50:00+00:00"Haghighi, Ahmad Reza"https://www.zbmath.org/authors/?q=ai:haghighi.ahmad-reza"Shahbazi Asl, Mohammad"https://www.zbmath.org/authors/?q=ai:shahbazi-asl.mohammadSummary: In this paper, the alternating segment Crank-Nicolson (nASCN) scheme is compared to the alternating group explicit-implicit (nAGEI) scheme for the dispersive equation with periodic boundary conditions. Both schemes are unconditionally stable and have a truncation error of the fourth-order in space. The comparison between the accuracy of these two schemes is presented in the numerical experiments.An improved PSO with detecting and local-learning strategy.https://www.zbmath.org/1453.902182021-02-27T13:50:00+00:00"Xia, Xuewen"https://www.zbmath.org/authors/?q=ai:xia.xuewen"Wei, Bo"https://www.zbmath.org/authors/?q=ai:wei.bo"Xie, Chengwang"https://www.zbmath.org/authors/?q=ai:xie.chengwangSummary: Particle swarm optimisation (PSO) has been applied to a variety of problems due to its simplicity of implement. However, the standard PSO suffers from premature convergence and slow global optimisation. This paper presents a novel PSO algorithm, in which detecting strategy and local-learning strategy are adopted to improve PSO's performance. In the new PSO algorithm, which is called DLPSO in this paper, search space of each dimension is divided into many equal subregions. According to statistical information of all particles' historical best position, the globally best particle can detect some inferior (or superior) subregions. In the local-learning strategy, the global best particle can carry out a local search during the later evolution process. The results of experiments show that the detecting strategy can act on the globally best particle to jump out of the likely local optimal solutions while local-learning strategy can help DLPSO obtain more accurate solutions. In addition, experimental results also demonstrate that DLPSO is more suitable for multimodal function optimisation while it has a comprehensive ability for function optimisation.Space-time finite element methods for parabolic evolution equations: discretization, a posteriori error estimation, adaptivity and solution.https://www.zbmath.org/1453.653442021-02-27T13:50:00+00:00"Steinbach, Olaf"https://www.zbmath.org/authors/?q=ai:steinbach.olaf"Yang, Huidong"https://www.zbmath.org/authors/?q=ai:yang.huidongSummary: In this work, we present an overview on the development of space-time finite element methods for the numerical solution of some parabolic evolution equations with the heat equation as a model problem. Instead of using more standard semidiscretization approaches, such as the method of lines or Rothe's method, our specific focus is on continuous space-time finite element discretizations in space and time simultaneously. Whereas such discretizations bring more flexibility to the space-time finite element error analysis and error control, they usually lead to higher computational complexity and memory consumptions in comparison with standard timestepping methods. Therefore, progress on a posteriori error estimation and respective adaptive schemes in the space-time domain is reviewed, which aims to save a number of degrees of freedom, and hence reduces complexity, and recovers optimal-order error estimates. Further, we provide a summary on recent advances in efficient parallel space-time iterative solution strategies for the related large-scale linear systems of algebraic equations that are crucial to make such all-at-once approaches competitive with traditional time-stepping methods. Finally, some numerical results are given to demonstrate the benefits of a particular adaptive space-time finite element method, the robustness of some space-time algebraic multigrid methods, and the applicability of space-time finite element methods for the solution of some parabolic optimal control problem.
For the entire collection see [Zbl 1425.65008].Extensions and analysis of worst-case parameter in weighted Jacobi's method for solving second order implicit PDEs.https://www.zbmath.org/1453.650712021-02-27T13:50:00+00:00"Kimmel, Gregory J."https://www.zbmath.org/authors/?q=ai:kimmel.gregory-j"Glatz, Andreas"https://www.zbmath.org/authors/?q=ai:glatz.andreasSummary: The optimal Jacobi parameter \((\omega)\) in Jacobi's iterative method is obtained for specific classes of matrices. We define \(\omega_{\text{opt}}\) as the worst-case optimal parameter. We show that matrices with nonzero elements only along the main diagonal and odd diagonals have \(\omega_{\text{opt}}=1\). We show \(\omega_{\text{opt}}\to 1\) holds for matrices with size \(n\) and nonzero diagonal \(d\) as \(n,d\to\infty\), where \(d\) is the distance from the main diagonal. Finally, we show an application which exploits these derived properties to reduce the number of required Jacobi iterations. This is especially useful for physical problems that involve 2nd order implicit PDEs (e.g. diffusion, fluids) with large sparse matrices, where a change in discretization can change which diagonals are nonzero.Kinematics and dynamics motion planning by polar piecewise interpolation and geometric considerations.https://www.zbmath.org/1453.700012021-02-27T13:50:00+00:00"Dupac, Mihai"https://www.zbmath.org/authors/?q=ai:dupac.mihaiSummary: The importance of numerical methods in science and engineering [\textit{S. C. Chapra} and \textit{R. P. Canale}, Numerical methods for engineers. 6. ed. New York, NY: McGraw-Hill (2010)] was long recognised and considered a fundamental factor in improving productivity and reducing production costs. The ability to model flexible systems and describe their trajectories [\textit{A. Gasparetto} et al., ``Trajectory planning in robotics'', Math. Comput. Sci. 6, 269--279 (2012; \url{doi:10.1007/s11786-012-0123-8})] involves usually the study of nonlinear coupled partial differential equations. Since their exact solutions are not normally feasible in practice, computational methods [\textit{V. Kumar} et al., ``Motion planning and control of robots'', in: Handbook of industrial robotics. 2nd ed. Hoboken, NJ: Wiley and sons. 295--315 (2007)] can be considered.
For the entire collection see [Zbl 1392.00002].Algorithms for solving linear, non-linear differential and differential-difference equations by approximating the solution to polynomials.https://www.zbmath.org/1453.340232021-02-27T13:50:00+00:00"Eisa, Sameh Abdelwahab Nasr"https://www.zbmath.org/authors/?q=ai:eisa.sameh-abdelwahab-nasrSummary: In this paper, a simple methodology for solving initial/boundary value problems for linear/non-linear differential and differential-difference equations is proposed, by approximating the solution to polynomials. The method is based on direct substitution by the assumed polynomials into DEs themselves and then solving system of equations to compute the coefficients of the polynomials. Comparisons of the results are performed with other works published during the last decade through practical examples, which have shown significant progress in the accuracy and the simplicity. This will give the researchers more advantage to use this method especially the engineers.Assessment of structural health monitoring by analyzing some modal parameters (I) (an inventory of methods and some developments).https://www.zbmath.org/1453.740752021-02-27T13:50:00+00:00"Rebenciuc, Mihai"https://www.zbmath.org/authors/?q=ai:rebenciuc.mihai"Bibic, Simona Mihaela"https://www.zbmath.org/authors/?q=ai:bibic.simona-mihaelaSummary: Structural health monitoring (SHM) evaluation consists to determining the modes (resonances) of vibration characteristic of the structure and each of them is represented by its modal parameters which can be obtained experimentally and can be analyzed by different procedures. In this regard, the present paper constitutes the first part of an extended paper and aims to inventory some methods of classical and non-classical mathematics with the specific computing scheme.
For the entire collection see [Zbl 1392.00002].Formalization and execution of linear algebra: from theorems to algorithms.https://www.zbmath.org/1453.682102021-02-27T13:50:00+00:00"Aransay, Jesús"https://www.zbmath.org/authors/?q=ai:aransay.jesus"Divasón, Jose"https://www.zbmath.org/authors/?q=ai:divason.joseSummary: In this work we present a formalization of the \textit{Rank Nullity} theorem of Linear Algebra in Isabelle/HOL. The formalization is of interest because of various reasons. First, it has been carried out based on the representation of mathematical structures proposed in the HOL Multivariate Analysis library of Isabelle/HOL (which is part of the standard distribution of the proof assistant). Hence, our proof shows the adequacy of such an infrastructure for the formalization of Linear Algebra. Moreover, we enrich the proof with an additional formalization of its \textit{computational} meaning; to this purpose, we choose to implement the Gauss-Jordan elimination algorithm for matrices over fields, prove it correct, and then apply the Isabelle code generation facility that permits to \textit{execute} the formalized algorithm. For the algorithm to be code generated, we use again the implementation of matrices available in the HOL Multivariate Analysis library, and enrich it with some necessary features. We report on the precise modifications that we introduce to get code execution from the original representation, and on the performance of the code obtained. We present an alternative verified type refinement of vectors that outperforms the original version. This refinement performs well enough as to be applied to the computation of the rank of some biomedical digital images. Our work proves itself as a suitable basis for the formalization of numerical Linear Algebra in HOL provers that can be successfully applied for computations of real case studies.
For the entire collection see [Zbl 1320.68017].A spectral approach for solving the nonclassical transport equation.https://www.zbmath.org/1453.821052021-02-27T13:50:00+00:00"Vasques, R."https://www.zbmath.org/authors/?q=ai:vasques.richard"Moraes, L. R. C."https://www.zbmath.org/authors/?q=ai:moraes.l-r-c"Barros, R. C."https://www.zbmath.org/authors/?q=ai:barros.rodrigo-c|barros.rui-c|de-barros.ricardo-c"Slaybaugh, R. N."https://www.zbmath.org/authors/?q=ai:slaybaugh.r-nSummary: This paper introduces a mathematical approach that allows one to numerically solve the nonclassical transport equation in a deterministic fashion using classical numerical procedures. The nonclassical transport equation describes particle transport for random statistically homogeneous systems in which the distribution function for free-paths between scattering centers is nonexponential. We use a spectral method to represent the nonclassical flux as a series of Laguerre polynomials in the free-path variable \(s\), resulting in a nonclassical equation that has the form of a classical transport equation. We present numerical results that validate the spectral approach, considering transport in slab geometry for both classical and nonclassical problems in the discrete ordinates formulation.Interval versions for special kinds of explicit linear multistep methods.https://www.zbmath.org/1453.651662021-02-27T13:50:00+00:00"Marciniak, Andrzej"https://www.zbmath.org/authors/?q=ai:marciniak.andrzej"Jankowska, Malgorzata A."https://www.zbmath.org/authors/?q=ai:jankowska.malgorzata-aSummary: In classical theory of explicit linear multistep methods there are known special kinds of methods which have less function evaluations (in comparison to other multistep methods) and, nevertheless, they give the same accuracy (order) of the approximations obtained. In this paper for such methods we propose their interval versions. It appears that enclosures to the exact solutions obtained by these methods are better in comparison to interval versions of other multistep methods with the same number of steps. The numerical examples presented show that sometimes these enclosures are even better than those obtained by interval methods based on high-order Taylor series.An optimally convergent higher-order finite element coupling method for interface and domain decomposition problems.https://www.zbmath.org/1453.654272021-02-27T13:50:00+00:00"Cheung, James"https://www.zbmath.org/authors/?q=ai:cheung.james"Gunzburger, Max"https://www.zbmath.org/authors/?q=ai:gunzburger.max-d"Bochev, Pavel"https://www.zbmath.org/authors/?q=ai:bochev.pavel-b"Perego, Mauro"https://www.zbmath.org/authors/?q=ai:perego.mauroSummary: We present a new, optimally accurate finite element method for interface problems that does not require matching interface grids or spatially coincident interfaces. The key idea is to enforce ``extended'' interface conditions through pullbacks onto the discretized interfaces. In so doing our approach circumvents the accuracy barriers prompted by polytopial approximations of the subdomains and enables high-order finite element solutions without needing more expensive curvilinear maps. Since the discrete interfaces are not required to match, the approach is also appropriate for multiphysics couplings where each subdomain is meshed independently and solved by a separate code. Error analysis reveals that the new approach is well posed and optimally convergent with respect to a broken \(H^1\) norm. Numerical examples confirm this result and also indicate optimal convergence in a broken \(L^2\) norm.Multi-agent list-based threshold-accepting algorithm for numerical optimisation.https://www.zbmath.org/1453.651342021-02-27T13:50:00+00:00"Lin, Juan"https://www.zbmath.org/authors/?q=ai:lin.juan"Zhong, Yiwen"https://www.zbmath.org/authors/?q=ai:zhong.yiwenSummary: Traditional list-based threshold-accepting (LBTA) algorithm is similar with simulated annealing (SA) algorithm, depends on an intense local search method, and utilises a list filling procedure with threshold values to search the solution space effectively. Inspired by the learning ability of particle swarm optimisation (PSO), multi-agent LBTA (MLBTA) involves the learning knowledge to guide its sampling, explores the solution space in a co-evolution mode. Compare with multi-agent SA (MSA) algorithm adapting the same local search version, MLBTA incorporates a dynamic list of threshold values which is adapted according to the topology of the solution space and tunes only one parameter. Dispense with sophisticated parameters as MSA, MLBTA balances the intensification and diversification iteratively. Computational results on functions optimisation and protein structure prediction (PSP) problems show that MLBTA algorithm achieves better or comparable performances with MSA.Convergence analysis on the Gibou-Min method for the Hodge projection.https://www.zbmath.org/1453.653882021-02-27T13:50:00+00:00"Yoon, Gangjoon"https://www.zbmath.org/authors/?q=ai:yoon.gangjoon"Park, Jea-Hyun"https://www.zbmath.org/authors/?q=ai:park.jea-hyun"Min, Chohong"https://www.zbmath.org/authors/?q=ai:min.chohongSummary: The Hodge projection of a vector field is the divergence-free component of its Helmholtz decomposition. In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou-Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition. In the decomposition by the Gibou-Min method, an important \(L^2\)-orthogonality holds between the gradient field and the solenoidal field, which is similar to the continuous Hodge decomposition. Using the orthogonality, we present a novel analysis which shows that the convergence order is \(1.5\) in the \(L^2\)-norm for approximating the divergence-free vector field. Numerical results are presented to validate our analyses.High-order quasi-compact difference schemes for fractional diffusion equations.https://www.zbmath.org/1453.652382021-02-27T13:50:00+00:00"Yu, Yanyan"https://www.zbmath.org/authors/?q=ai:yu.yanyan"Deng, Weihua"https://www.zbmath.org/authors/?q=ai:deng.weihua"Wu, Yujiang"https://www.zbmath.org/authors/?q=ai:wu.yujiangSummary: The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has a space fractional derivative, which characterizes Lévy flights. Sometimes the infinite variance of Lévy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more `physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high-order quasi-compact discretizations for the space fractional derivative and the tempered space fractional derivative. The fourth-order quasi-compact discretization for the space fractional derivative is applied to solve a space fractional diffusion equation, and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart, the fourth-order quasi-compact scheme, and the convergence orders are verified numerically.A direct method for nonlinear ill-posed problems.https://www.zbmath.org/1453.651252021-02-27T13:50:00+00:00"Lakhal, A."https://www.zbmath.org/authors/?q=ai:lakhal.arefAuthor's abstract: We propose a direct method for solving nonlinear ill-posed problems in Banach-spaces. The method is based on a stable inversion formula we explicitly compute by applying techniques for analytic functions. Furthermore, we investigate the convergence and stability of the method and prove that the derived noniterative algorithm is a regularization. The inversion formula provides a systematic sensitivity analysis. The approach is applicable to a wide range of nonlinear ill-posed problems. We test the algorithm on a nonlinear problem of travel-time inversion in seismic tomography. Numerical results illustrate the robustness and efficiency of the algorithm.
Reviewer: Xiaolong Qin (Chengdu)Discontinuous Legendre wavelet Galerkin method for the generalised Burgers-Fisher equation.https://www.zbmath.org/1453.653722021-02-27T13:50:00+00:00"Zheng, Xiaoyang"https://www.zbmath.org/authors/?q=ai:zheng.xiaoyang"Fu, Yong"https://www.zbmath.org/authors/?q=ai:fu.yong"Wei, Zhengyuan"https://www.zbmath.org/authors/?q=ai:wei.zhengyuanSummary: This paper presents discontinuous Legendre wavelet Galerkin (DLWG) technique for solving the generalised Burgers-Fisher equation. Weak formulation of this equation and numerical fluxes are addressed by utilising the advantages of the both Legendre wavelet and discontinuous Galerkin (DG) approach. Finally, numerical experiments demonstrate the validity and utility of the DLWG method.Automatic alignment for three-dimensional tomographic reconstruction.https://www.zbmath.org/1453.654662021-02-27T13:50:00+00:00"van Leeuwen, Tristan"https://www.zbmath.org/authors/?q=ai:van-leeuwen.tristan"Maretzke, Simon"https://www.zbmath.org/authors/?q=ai:maretzke.simon"Batenburg, K. Joost"https://www.zbmath.org/authors/?q=ai:batenburg.kees-joostHigh-order discretization of a stable time-domain integral equation for 3D acoustic scattering.https://www.zbmath.org/1453.654472021-02-27T13:50:00+00:00"Barnett, Alex"https://www.zbmath.org/authors/?q=ai:barnett.alex-h"Greengard, Leslie"https://www.zbmath.org/authors/?q=ai:greengard.leslie-f"Hagstrom, Thomas"https://www.zbmath.org/authors/?q=ai:hagstrom.thomas-mSummary: We develop a high-order, explicit method for acoustic scattering in three space dimensions based on a combined-field time-domain integral equation. The spatial discretization, of Nyström type, uses Gaussian quadrature on panels combined with a special treatment of the weakly singular kernels arising in near-neighbor interactions. In time, a new class of convolution splines is used in a predictor-corrector algorithm. Experiments on a torus and a perturbed torus are used to explore the stability and accuracy of the proposed scheme. This involved around one thousand solver runs, at up to 8th order and up to around 20,000 spatial unknowns, demonstrating 5--9 digits of accuracy. In addition we show that parameters in the combined field formulation, chosen on the basis of analysis for the sphere and other convex scatterers, work well in these cases.Boundary observability of semi-discrete second-order integro-differential equations derived from piecewise Hermite cubic orthogonal spline collocation method.https://www.zbmath.org/1453.654652021-02-27T13:50:00+00:00"Xu, Da"https://www.zbmath.org/authors/?q=ai:xu.daThe author studies the boundary observability of semi-discrete second-order integro-differential equations derived from a piecewise Hermite cubic orthogonal spline collocation method. First, the author analyzes the spectrum of the equation considered and gives a representation of the solution with a new finite sequence to the associated system. Then, he proves that the new sequence in the class \(C_{h}\) satisfies the theorems stated. Piecewise Hermite cubic orthogonal spline collocation semi-discretization is considered.
Reviewer: Seenith Sivasundaram (Daytona Beach)Computational method for one-dimensional heat equation subject to non-local conditions.https://www.zbmath.org/1453.653552021-02-27T13:50:00+00:00"Behroozifar, Mahmoud"https://www.zbmath.org/authors/?q=ai:behroozifar.mahmoudSummary: In this paper, a method based on Bernstein polynomial is proposed for numerical solution of an one-dimensional heat equation subject to non-local boundary conditions. Properties of Bernstein polynomial and operational matrices of integration, differentiation and the product are stated and utilised to reduce the solution of the given elliptic partial differential equation to the solution of the system of algebraic equations. Illustrative numerical tests are given to compare the efficiency and accuracy of the new method with other methods.Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives.https://www.zbmath.org/1453.651602021-02-27T13:50:00+00:00"Odibat, Zaid"https://www.zbmath.org/authors/?q=ai:odibat.zaid-m"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: We introduce a new generalized Caputo-type fractional derivative which generalizes Caputo fractional derivative. Some characteristics were derived to display the new generalized derivative features. Then, we present an adaptive predictor corrector method for the numerical solution of generalized Caputo-type initial value problems. The proposed algorithm can be considered as a fractional extension of the classical Adams-Bashforth-Moulton method. Dynamic behaviors of some fractional derivative models are numerically discussed. We believe that the presented generalized Caputo-type fractional derivative and the proposed algorithm are expected to be further used to formulate and simulate many generalized Caputo type fractional models.Some results on the regularization of LSQR for large-scale discrete ill-posed problems.https://www.zbmath.org/1453.650802021-02-27T13:50:00+00:00"Huang, Yi"https://www.zbmath.org/authors/?q=ai:huang.yi|huang.yi.1"Jia, ZhongXiao"https://www.zbmath.org/authors/?q=ai:jia.zhongxiaoThe authors derive bounds for the \(2\)-norm distance between the \(k\)-dimensional Krylov subspace of the LSQR algorithm and the \(k\)-dimensional right singular space. Therefore they show that the algorithm has better regularizing effects for severely and moderately ill-posed problems than for mildly ill-posed ones. Numerical experiments illustrating these regularizing properties of LSQR algorithm are also presented.
Reviewer: Constantin Popa (Constanţa)A moment-of-fluid method for diffusion equations on irregular domains in multi-material systems.https://www.zbmath.org/1453.652562021-02-27T13:50:00+00:00"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.1|liu.yang.13|liu.yang.23|liu.yang.19|liu.yang.3|liu.yang.9|liu.yang.16|liu.yang.8|liu.yang.20|liu.yang.7|liu.yang.11|liu.yang.2|liu.yang.14|liu.yang.12|liu.yang.18|liu.yang.22|liu.yang.21|liu.yang.10|liu.yang.6|liu.yang.17|liu.yang.5|liu.yang.4|liu.yang.15|liu.yang"Sussman, Mark"https://www.zbmath.org/authors/?q=ai:sussman.mark"Lian, Yongsheng"https://www.zbmath.org/authors/?q=ai:lian.yongsheng"Yousuff Hussaini, M."https://www.zbmath.org/authors/?q=ai:hussaini.m-yousuffSummary: A new numerical method is developed for the solution of the diffusion problem in a system of several materials. In such a system, the diffusion coefficients are piecewise continuous and jumps in their values can occur across the complex-shaped interfaces between contiguous materials. The boundary conditions along the complex-shaped interfaces can either be a jump condition boundary condition, a Neumann boundary condition, or a Dirichlet boundary condition. The moment-of-fluid (MOF) procedure is employed to reconstruct the interfaces. This procedure enables accurate reconstruction of any number of material interfaces in a computational cell. Furthermore, MOF is a volume preserving reconstruction, as well as capable of capturing thin filamentary regions without the necessity of adaptive mesh refinement. The proposed method is tested on multi-material diffusion problems which demonstrates its potential to enable numerical simulation of complex flows of technological importance relevant to predicting the heat transfer rate in materials and manufacturing processes. Results using the new method are reported on problems in complex (filamentary) domains, and it is found that the method is very efficient at approximating the temperature, the temperature gradient, and the interfacial heat flux, as compared to traditional approaches.Ghost solutions with centered schemes for one-dimensional transport equations with Neumann boundary conditions.https://www.zbmath.org/1453.652202021-02-27T13:50:00+00:00"Inglard, Mélanie"https://www.zbmath.org/authors/?q=ai:inglard.melanie"Lagoutière, Frédéric"https://www.zbmath.org/authors/?q=ai:lagoutiere.frederic"Rugh, Hans Henrik"https://www.zbmath.org/authors/?q=ai:rugh.hans-henrikSummary: This paper is devoted to the space-centered \(\theta \)-scheme for the transport equation with constant velocity on an interval, with homogeneous Neumann boundary data on each side. The numerical solution presents a weird behavior, as if the initial datum was periodical over \(\mathbb{R} \), with a period that would be twice or four times (depending on the case) the length of the considered interval. We precisely formulate this statement and prove it. We also study a similar behavior in the case of Dirichlet boundary conditions.Stability analysis of split-explicit free surface ocean models: implication of the depth-independent barotropic mode approximation.https://www.zbmath.org/1453.860172021-02-27T13:50:00+00:00"Demange, Jérémie"https://www.zbmath.org/authors/?q=ai:demange.jeremie"Debreu, Laurent"https://www.zbmath.org/authors/?q=ai:debreu.laurent"Marchesiello, Patrick"https://www.zbmath.org/authors/?q=ai:marchesiello.patrick"Lemarié, Florian"https://www.zbmath.org/authors/?q=ai:lemarie.florian"Blayo, Eric"https://www.zbmath.org/authors/?q=ai:blayo.eric"Eldred, Christopher"https://www.zbmath.org/authors/?q=ai:eldred.christopherSummary: The evolution of the oceanic free-surface is responsible for the propagation of fast surface gravity waves, which approximatively propagate at speed \(\sqrt{g H}\) (with \(g\) the gravity and \(H\) the local water depth). In the deep ocean, this phase speed is roughly two orders of magnitude faster than the fastest internal gravity waves. The very strong stability constraint imposed by those fast surface waves on the time-step of numerical models is handled using a mode splitting between slow (internal/baroclinic) and fast (external/barotropic) motions to allow the possibility to adopt specific numerical treatments in each component. The barotropic mode is traditionally approximated by the vertically integrated flow because it has only slight vertical variations. However the implications of this assumption on the stability of the splitting are not well documented. In this paper, we describe a stability analysis of the mode splitting technique based on an eigenvector decomposition using the true (depth-dependent) barotropic mode. This allows us to quantify the amount of dissipation required to stabilize the approximative splitting. We show that, to achieve stable integrations, the dissipation usually applied through averaging filters can be drastically reduced when incorporated at the level of the barotropic time stepping. The benefits are illustrated by numerical experiments. In addition, the formulation of a new mode splitting algorithm using the depth-dependent barotropic mode is introduced.A fast method for solving a block tridiagonal quasi-Toeplitz linear system.https://www.zbmath.org/1453.650582021-02-27T13:50:00+00:00"Belhaj, Skander"https://www.zbmath.org/authors/?q=ai:belhaj.skander"Hcini, Fahd"https://www.zbmath.org/authors/?q=ai:hcini.fahd"Zhang, Yulin"https://www.zbmath.org/authors/?q=ai:zhang.yulinSummary: This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by [\textit{L. Du} et al., Appl. Math. Lett. 75, 74--81 (2018; Zbl 1377.65037)], we propose a more general algorithm for such systems. The algorithm is based on a block decomposition for block tridiagonal quasi-Toeplitz matrices and the Sherman-Morrison-Woodbury inversion formula. We also compare the proposed approach to the standard block \(LU\) decomposition method and the Gauss algorithm. A theoretical error analysis is also presented. All algorithms have been implemented in Matlab. Numerical experiments performed on a wide variety of test problems show the effectiveness of our algorithm in terms of efficiency, stability and robustness.Modified minimal error method for nonlinear ill-posed problems.https://www.zbmath.org/1453.651232021-02-27T13:50:00+00:00"Sabari, M."https://www.zbmath.org/authors/?q=ai:sabari.m"George, Santhosh"https://www.zbmath.org/authors/?q=ai:george.santhoshSummary: An error estimate for the minimal error method for nonlinear ill-posed problems under general a Hölder-type source condition is not known. We consider a modified minimal error method for nonlinear ill-posed problems. Using a Hölder-type source condition, we obtain an optimal order error estimate. We also consider the modified minimal error method with noisy data and provide an error estimate.An efficient fifth-order Steffensen-type method for solving systems of nonlinear equations.https://www.zbmath.org/1453.651012021-02-27T13:50:00+00:00"Singh, Anuradha"https://www.zbmath.org/authors/?q=ai:singh.anuradhaSummary: In this paper, we present a three-step Steffensen-type iterative method of order five for solving systems of nonlinear equations. Various particular cases of the proposed method are considered. The general form of computational efficiency of the proposed scheme is compared to existing techniques. Numerical examples are given to show the performance of the proposed method with some existing schemes. We observed from the comparison of the new scheme with some known methods that the proposed scheme shows high efficiency index than others.Multiphase-field modelling of crack propagation in geological materials and porous media with Drucker-Prager plasticity.https://www.zbmath.org/1453.860082021-02-27T13:50:00+00:00"Späth, Michael"https://www.zbmath.org/authors/?q=ai:spath.michael"Herrmann, Christoph"https://www.zbmath.org/authors/?q=ai:herrmann.christoph"Prajapati, Nishant"https://www.zbmath.org/authors/?q=ai:prajapati.nishant"Schneider, Daniel"https://www.zbmath.org/authors/?q=ai:schneider.daniel"Schwab, Felix"https://www.zbmath.org/authors/?q=ai:schwab.felix"Selzer, Michael"https://www.zbmath.org/authors/?q=ai:selzer.michael"Nestler, Britta"https://www.zbmath.org/authors/?q=ai:nestler.brittaSummary: A multiphase-field approach for elasto-plastic and anisotropic brittle crack propagation in geological systems consisting of different regions of brittle and ductile materials is presented and employed to computationally study crack propagation. Plastic deformation in elasto-plastic materials such as frictional, granular or porous materials is modelled with the pressure-sensitive Drucker-Prager plasticity model. This plasticity model is combined with a multiphase-field model fulfilling the mechanical jump conditions in diffuse solid-solid interfaces. The validity of the plasticity model with phase-inherent stress and strain fields is shown, in comparison with sharp interface finite element solutions. The proposed model is capable of simulating crack formation in heterogeneous multiphase systems comprising both purely elastic and inelastic phases. We investigate the influence of different material parameters on the crack propagation with tensile tests in single- and two-phase materials. To show the applicability of the model, crack propagation in a multiphase domain with brittle and elasto-plastic components is performed.Robust modeling of hysteretic capillary pressure and relative permeability for two phase flow in porous media.https://www.zbmath.org/1453.762092021-02-27T13:50:00+00:00"Yoon, Hyun C."https://www.zbmath.org/authors/?q=ai:yoon.hyun-c"Zhou, Peng"https://www.zbmath.org/authors/?q=ai:zhou.peng.1|zhou.peng"Kim, Jihoon"https://www.zbmath.org/authors/?q=ai:kim.jihoonSummary: We investigate a robust and systematic modeling approach for hysteretic capillary pressure and relative permeability in porous media by using the theory of plasticity, considering that plasticity and hysteresis exhibit both irreversible physical processes. Focusing on the immiscible two-phase flow, we investigate stability analysis and find that the method based on the plasticity can yield well posedness (contractivity) and algorithmic stability (B-stability). This modeling approach can track and compute history-dependent flow properties such as residual saturation. In numerical simulation, we apply the algorithm of the 1D isotropic/kinematic hardening plasticity to reservoir simulation of gas-water flow. For weak and strong capillarity, the modeling yields strong numerical stability even for several drainage-imbibition processes. We also identify differences between with and without hysteresis, showing the importance of hysteretic capillary pressure and relative permeability. Thus, the hysteresis modeling based on the theory of plasticity is promising for robust numerical simulation of strong hysteresis.Numerical solutions of the dissipative nonlinear Schrödinger equation with variable coefficient arises in elastic tube.https://www.zbmath.org/1453.652342021-02-27T13:50:00+00:00"Tay, Kim Gaik"https://www.zbmath.org/authors/?q=ai:tay.kim-gaik"Choy, Yan Yee"https://www.zbmath.org/authors/?q=ai:choy.yan-yee"Tiong, Wei King"https://www.zbmath.org/authors/?q=ai:tiong.wei-king"Ong, Chee Tiong"https://www.zbmath.org/authors/?q=ai:ong.cheetiongSummary: In this paper, we solved the dissipative nonlinear Schrödinger Equation (DNLSV) with variable coefficient by Crank-Nicolson (CN) implicit finite-difference method. The DNLSV equation arises in nonlinear wave modulation in an elastic tube with a symmetrical stenosis filled with viscous fluid. We then compared numerical solutions with its progressive wave solution. The CN scheme is consistent with the differential equation and is unconditionally stable.The fully-implicit finite difference method for solving nonlinear inverse parabolic problems with unknown source term.https://www.zbmath.org/1453.652912021-02-27T13:50:00+00:00"Mazraeh, Hassan Dana"https://www.zbmath.org/authors/?q=ai:mazraeh.hassan-dana"Pourgholi, Reza"https://www.zbmath.org/authors/?q=ai:pourgholi.reza"Tavana, Sahar"https://www.zbmath.org/authors/?q=ai:tavana.saharSummary: A numerical procedure based on a fully implicit finite difference method for an inverse problem of identification of an unknown source in a heat equation is presented. The approach of the proposed method is to approximate unknown function from the solution of the minimisation problem based on the overspecified data. This problem is ill-posed, in the sense that the solution (if it exist) does not depend continuously on the data. To regularise this ill-conditioned, we apply the Tikhonov regularisation 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. A stability analysis shows that this numerical scheme approximation is unconditionally stable. Numerical results for two inverse source identification problems show that the proposed numerical algorithm is simple, accurate, stable and computationally efficient.A regularization interpretation of the proximal point method for weakly convex functions.https://www.zbmath.org/1453.901202021-02-27T13:50:00+00:00"Hoheisel, Tim"https://www.zbmath.org/authors/?q=ai:hoheisel.tim"Laborde, Maxime"https://www.zbmath.org/authors/?q=ai:laborde.maxime"Oberman, Adam"https://www.zbmath.org/authors/?q=ai:oberman.adam-mSummary: Empirical evidence and theoretical results suggest that the proximal point method can be computed approximately and still converge faster than the corresponding gradient descent method, in both the stochastic and exact gradient case. In this article we provide a perspective on this result by interpreting the method as gradient descent on a regularized function. This perspective applies in the case of weakly convex functions where proofs of the faster rates are not available. Using this analysis we find the optimal value of the regularization parameter in terms of the weak convexity.A bi-fidelity method for the multiscale Boltzmann equation with random parameters.https://www.zbmath.org/1453.653602021-02-27T13:50:00+00:00"Liu, Liu"https://www.zbmath.org/authors/?q=ai:liu.liu"Zhu, Xueyu"https://www.zbmath.org/authors/?q=ai:zhu.xueyuSummary: In this paper, we study the multiscale Boltzmann equation with multi-dimensional random parameters by a bi-fidelity stochastic collocation (SC) method developed in [\textit{A. Narayan} et al., SIAM J. Sci. Comput. 36, No. 2, 495--521 (2014; Zbl 1296.65013); \textit{X. Zhu} et al., SIAM/ASA J. Uncertain. Quantif. 2, 444--463 (2014; Zbl 1306.65010); ``A multi-fidelity collocation method for time-dependent parameterized problems'', AIAA SciTech Forum (2017)]. By choosing the compressible Euler system as the low-fidelity model, we adapt the bi-fidelity SC method to combine computational efficiency of the low-fidelity model with high accuracy of the high-fidelity (Boltzmann) model. With only a small number of high-fidelity asymptotic-preserving solver runs for the Boltzmann equation, the bi-fidelity approximation can capture well the macroscopic quantities of the solution to the Boltzmann equation in the random space. A priori estimate on the accuracy between the high- and bi-fidelity solutions together with a convergence analysis is established. Finally, we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method.Second derivative two-step collocation methods for ordinary differential equations.https://www.zbmath.org/1453.651922021-02-27T13:50:00+00:00"Fazeli, S."https://www.zbmath.org/authors/?q=ai:fazeli.somayyeh"Hojjati, G."https://www.zbmath.org/authors/?q=ai:hojjati.gholamrezaSummary: We introduce second derivative two-step collocation methods for the numerical integration of ordinary differential equations. In these methods, the solution of the problem in each step depends on the numerical solution in some points in the two previous steps. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability properties. The construction technique, analysis of the order of accuracy and linear stability properties of the methods are described.Enhanced Box-Muller method for high quality Gaussian random number generation.https://www.zbmath.org/1453.650122021-02-27T13:50:00+00:00"Addaim, Adnane"https://www.zbmath.org/authors/?q=ai:addaim.adnane"Gretete, Driss"https://www.zbmath.org/authors/?q=ai:gretete.driss"Madi, Abdessalam Ait"https://www.zbmath.org/authors/?q=ai:madi.abdessalam-aitSummary: Fast and high-quality Gaussian random number generation (GRNG) is a key capability for simulations across a wide range of disciplines. In this article, we present an enhanced Box-Muller method for GRNG using one uniform variable. Its probability density function (PDF) is given in closed form as a function of one parameter. In this article, the theoretical basis of this method is quite thoroughly discussed and is evaluated using several different statistical tests, including the chi-square test and the Anderson-Darling test. The simulations results show good performances of this method which generates accurately a true Gaussian PDF even at very high \(\sigma\) (standard deviations) values in comparison with the standard Box-Muller method.Maximum-likelihood estimation of the random-clumped multinomial model as a prototype problem for large-scale statistical computing.https://www.zbmath.org/1453.623982021-02-27T13:50:00+00:00"Raim, Andrew M."https://www.zbmath.org/authors/?q=ai:raim.andrew-m"Gobbert, Matthias K."https://www.zbmath.org/authors/?q=ai:gobbert.matthias-k"Neerchal, Nagaraj K."https://www.zbmath.org/authors/?q=ai:neerchal.nagaraj-k"Morel, Jorge G."https://www.zbmath.org/authors/?q=ai:morel.jorge-gSummary: Numerical methods are needed to obtain maximum-likelihood estimates (MLEs) in many problems. Computation time can be an issue for some likelihoods even with modern computing power. We consider one such problem where the assumed model is a random-clumped multinomial distribution. We compute MLEs for this model in parallel using the Toolkit for Advanced Optimization software library. The computations are performed on a distributed-memory cluster with low latency interconnect. We demonstrate that for larger problems, scaling the number of processes improves wall clock time significantly. An illustrative example shows how parallel MLE computation can be useful in a large data analysis. Our experience with a direct numerical approach indicates that more substantial gains may be obtained by making use of the specific structure of the random-clumped model.A preconditioned multiple shooting shadowing algorithm for the sensitivity analysis of chaotic systems.https://www.zbmath.org/1453.490132021-02-27T13:50:00+00:00"Shawki, Karim"https://www.zbmath.org/authors/?q=ai:shawki.karim"Papadakis, George"https://www.zbmath.org/authors/?q=ai:papadakis.georgeSummary: We propose a preconditioner that can accelerate the rate of convergence of the Multiple Shooting Shadowing (MSS) method [\textit{P. J. Blonigan} and \textit{Q. Wang}, J. Comput. Phys. 354, 447--475 (2018; Zbl 1380.37052)]. This recently proposed method can be used to compute derivatives of time-averaged objectives (also known as sensitivities) to system parameter(s) for chaotic systems. We propose a block diagonal preconditioner, which is based on a partial singular value decomposition of the MSS constraint matrix. The preconditioner can be computed using matrix-vector products only (i.e. it is matrix-free) and is fully parallelised in the time domain. Two chaotic systems are considered, the Lorenz system and the 1D Kuramoto Sivashinsky equation. Combination of the preconditioner with a regularisation method leads to tight bracketing of the eigenvalues to a narrow range. This combination results in a significant reduction in the number of iterations, and renders the convergence rate almost independent of the number of degrees of freedom of the system, and the length of the trajectory that is used to compute the time-averaged objective. This can potentially allow the method to be used for large chaotic systems (such as turbulent flows) and optimal control applications. The singular value decomposition of the constraint matrix can also be used to quantify the effect of the system condition on the accuracy of the sensitivities. In fact, neglecting the singular modes affected by noise, we recover the correct values of sensitivity that match closely with those obtained with finite differences for the Kuramoto Sivashinsky equation in the light turbulent regime. We notice a similar improvement for the Lorenz system as well.Symbolic-numerical optimization and realization of the method of collocations and least residuals for solving the Navier-Stokes equations.https://www.zbmath.org/1453.760372021-02-27T13:50:00+00:00"Shapeev, Vasily P."https://www.zbmath.org/authors/?q=ai:shapeev.vasilii-pavlovich"Vorozhtsov, Evgenii V."https://www.zbmath.org/authors/?q=ai:vorozhtsov.evgenii-vasilevichSummary: The computer algebra system (CAS) Mathematica has been applied for constructing the optimal iteration processes of the Gauss-Seidel type at the solution of PDE's by the method of collocations and least residuals. The possibilities of the proposed approaches are shown by the examples of the solution of boundary-value problems for the 2D Navier-Stokes equations.
For the entire collection see [Zbl 1346.68010].A new method based on artificial neural networks for solving general nonlinear systems.https://www.zbmath.org/1453.651042021-02-27T13:50:00+00:00"Abbasnejad, H."https://www.zbmath.org/authors/?q=ai:abbasnejad.h"Jafarian, A."https://www.zbmath.org/authors/?q=ai:jafarian.amin|jafarian.ali-a|jafarian.ahmad|jafarian.amiri-seyyed-majidSummary: Implementation of the amazing features of the human brain in an artificial system has long been considered. It seems that simulating the human nervous system is a recent development in applied mathematics and computer sciences. The objective of this research is to introduce an efficient iterative method based on artificial neural networks for numerically solving nonlinear algebraic systems of polynomial equations. The method first performs some simple algebraic manipulations to convert the origin system to an approximated unconstrained optimisation problem. Subsequently, the resulting nonlinear minimisation problem is solved iteratively using the neural networks approach. For this aim, a suitable five-layer feed-back neural architecture is formed and trained using a back-propagation supervised learning algorithm which is based on the gradient descent rule. Ultimately, some numerical examples with comparisons are given to demonstrate the high accuracy and the ease of implementation of the present technique over other classical methods.Particle smoothing via Markov chain Monte Carlo in general state space models.https://www.zbmath.org/1453.650262021-02-27T13:50:00+00:00"Gao, Meng"https://www.zbmath.org/authors/?q=ai:gao.meng"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui.4|zhang.hui.8|zhang.hui.5|zhang.hui.7|zhang.hui.9|zhang.hui|zhang.hui.1|zhang.hui.10|zhang.hui.3|zhang.hui.2|zhang.hui.6|zhang.hui.11Summary: Sequential Monte Carlo (SMC) methods (also known as particle filter) provide a way to solve the state estimation problem in nonlinear non-Gaussian state space models (SSM) through numerical approximation. Particle smoothing is one retrospective state estimation method based on particle filtering. In this paper, we propose a new particle smoother. The basic idea is easy and leads to a forward-backward procedure, where the Metropolis-Hastings algorithm is used to resample the filtering particles. The goodness of the new scheme is assessed using a nonlinear SSM. It is concluded that this new particle smoother is suitable for state estimation in complicated dynamical systems.An efficient numerical method for the solution of third order boundary value problem in ordinary differential equations.https://www.zbmath.org/1453.651742021-02-27T13:50:00+00:00"Pandey, Pramod Kumar"https://www.zbmath.org/authors/?q=ai:kumar-pandey.pramodSummary: In this article we have considered linear third order boundary value problems and proposed an efficient difference method for numerical solution of the problems. We have shown that proposed method is convergent and second order accurate. The numerical results in experiment on some test problems show the simplicity and efficiency of the method.Convergence results for implicit-explicit general linear methods.https://www.zbmath.org/1453.651672021-02-27T13:50:00+00:00"Sandu, Adrian"https://www.zbmath.org/authors/?q=ai:sandu.adrianSummary: This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebraic equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with the full theoretical orders under general assumptions. The convergence results require the time steps to be sufficiently small, with upper bounds that are independent on the stiffness of the problem.On multiple eigenvalues of a matrix dependent on a parameter.https://www.zbmath.org/1453.150082021-02-27T13:50:00+00:00"Kalinina, Elizabeth A."https://www.zbmath.org/authors/?q=ai:kalinina.elizabeth-aSummary: In this paper, a square matrix with elements linearly dependent on a parameter is considered. We propose an algorithm to find all the values of the parameter such that the matrix has a multiple eigenvalue. We construct a polynomial whose roots are these values of the parameter. A numerical example shows how the algorithm works.
For the entire collection see [Zbl 1346.68010].A note on a posteriori error bounds for numerical solutions of elliptic equations with a piecewise constant reaction coefficient having large jumps.https://www.zbmath.org/1453.654112021-02-27T13:50:00+00:00"Korneev, V. G."https://www.zbmath.org/authors/?q=ai:korneev.vadim-glebovichSummary: We have derived guaranteed, robust, and fully computable a posteriori error bounds for approximate solutions of the equation \(\Delta \Delta u + \Bbbk^2 u = f\), where the coefficient \(\Bbbk \geqslant 0\) is a constant in each subdomain (finite element) and chaotically varies between subdomains in a sufficiently wide range. For finite element solutions, these bounds are robust with respect to \(\Bbbk \in [0,ch^{-2}]\), \(c = \text{const}\), and possess some other good features. The coefficients in front of two typical norms on their right-hand sides are only insignificantly worse than those obtained earlier for \(\Bbbk \equiv\text{const}\). The bounds can be calculated without resorting to the equilibration procedures, and they are sharp for at least low-order methods. The derivation technique used in this paper is similar to the one used in our preceding papers (2017--2019) for obtaining a posteriori error bounds that are not improvable in the order of accuracy.Symbolic-numeric algorithms for solving BVPs for a system of ODEs of the second order: multichannel scattering and eigenvalue problems.https://www.zbmath.org/1453.651872021-02-27T13:50:00+00:00"Gusev, A. A."https://www.zbmath.org/authors/?q=ai:gusev.alexander-a"Gerdt, V. P."https://www.zbmath.org/authors/?q=ai:gerdt.vladimir-p"Hai, L. L."https://www.zbmath.org/authors/?q=ai:hai.lili"Derbov, V. L."https://www.zbmath.org/authors/?q=ai:derbov.vladimir-l"Vinitsky, S. I."https://www.zbmath.org/authors/?q=ai:vinitsky.sergue-i"Chuluunbaatar, O."https://www.zbmath.org/authors/?q=ai:chuluunbaatar.ochbadrakhSummary: Symbolic-numeric algorithms for solving multichannel scattering and eigenvalue problems of the waveguide or tunneling type for systems of ODEs of the second order with continuous and piecewise continuous coefficients on an axis are presented. The boundary-value problems are formulated and discretized using the FEM on a finite interval with interpolating Hermite polynomials that provide the required continuity of the derivatives of the approximated solutions. The accuracy of the approximate solutions of the boundary-value problems, reduced to a finite interval, is checked by comparing them with the solutions of the original boundary-value problems on the entire axis, which are calculated by matching the fundamental solutions of the ODE system. The efficiency of the algorithms implemented in the computer algebra system Maple is demonstrated by calculating the resonance states of a multichannel scattering problem on the axis for clusters of a few identical particles tunneling through Gaussian barriers.
For the entire collection see [Zbl 1346.68010].Efficient simplification techniques for special real quantifier elimination with applications to the synthesis of optimal numerical algorithms.https://www.zbmath.org/1453.120022021-02-27T13:50:00+00:00"Eraşcu, Mădălina"https://www.zbmath.org/authors/?q=ai:erascu.madalinaSummary: This paper presents efficient simplification techniques tailored for sign semi-definite conditions (SsDCs). The SsDCs for a polynomial \(f\in \mathbb {R}[y]\) with parametric coefficients are written as \(\substack {\forall \\ y \\ {L\leq y \leq U}} f(y) \geq 0\) and \(\substack {\forall \\ y \\ {L\leq y \leq U}} f(y) \leq 0\). We give sufficient conditions for the simplification techniques to be sound for linear and quadratic polynomials. We show their effectiveness compared to state of the art quantifier elimination tools for input formulae occurring in the optimal numerical algorithms synthesis problem by an implementation on top of \(\mathtt {Reduce}\) command of Mathematica.
For the entire collection see [Zbl 1346.68010].The complexity of cylindrical algebraic decomposition with respect to polynomial degree.https://www.zbmath.org/1453.130792021-02-27T13:50:00+00:00"England, Matthew"https://www.zbmath.org/authors/?q=ai:england.matthew"Davenport, James H."https://www.zbmath.org/authors/?q=ai:davenport.james-haroldSummary: Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be improved by adapting to take advantage of any equational constraints (ECs): equations logically implied by the input. Intuitively, we expect the double exponent in the complexity to decrease by one for each EC. In ISSAC 2015 the present authors [Proceedings of the 40th international symposium on symbolic and algebraic computation. New York, NY: ACM, 165--172 (2015; Zbl 1346.68283)] proved this for the factor in the complexity bound dependent on the number of polynomials in the input. However, the other term, that dependent on the degree of the input polynomials, remained unchanged.{
} In the present paper the authors investigate how CAD in the presence of ECs could be further refined using the technology of Gröbner bases to move towards the intuitive bound for polynomial degree.
For the entire collection see [Zbl 1346.68010].A numerical method for computing border curves of bi-parametric real polynomial systems and applications.https://www.zbmath.org/1453.650392021-02-27T13:50:00+00:00"Chen, Changbo"https://www.zbmath.org/authors/?q=ai:chen.changbo"Wu, Wenyuan"https://www.zbmath.org/authors/?q=ai:wu.wenyuanSummary: For a bi-parametric real polynomial system with parameter values restricted to a finite rectangular region, under certain assumptions, we introduce the notion of border curve. We propose a numerical method to compute the border curve, and provide a numerical error estimation.{
} The border curve enables us to construct a so-called ``solution map''. For a given value \(u\) of the parameters inside the rectangle but not on the border, the solution map tells the subset that \(u\) belongs to together with a connected path from the corresponding sample point \(w\) to \(u\). Consequently, all the real solutions of the system at \(u\) (which are isolated) can be obtained by tracking a real homotopy starting from all the real roots at \(w\) throughout the path. The effectiveness of the proposed method is illustrated by some examples.
For the entire collection see [Zbl 1346.68010].Sparse Gaussian elimination modulo \(p\): an update.https://www.zbmath.org/1453.650862021-02-27T13:50:00+00:00"Bouillaguet, Charles"https://www.zbmath.org/authors/?q=ai:bouillaguet.charles"Delaplace, Claire"https://www.zbmath.org/authors/?q=ai:delaplace.claireSummary: This paper considers elimination algorithms for sparse matrices over finite fields. We mostly focus on computing the rank, because it raises the same challenges as solving linear systems, while being slightly simpler.{
} We developed a new sparse elimination algorithm inspired by the Gilbert-Peierls sparse LU factorization, which is well-known in the numerical computation community. We benchmarked it against the usual right-looking sparse Gaussian elimination and the Wiedemann algorithm using the sparse integer matrix collection of Jean-Guillaume Dumas.{
} We obtain large speedups (\(1000\times\) and more) on many cases. In particular, we are able to compute the rank of several large sparse matrices in seconds or minutes, compared to days with previous methods.
For the entire collection see [Zbl 1346.68010].Algorithmic computation of polynomial amoebas.https://www.zbmath.org/1453.141452021-02-27T13:50:00+00:00"Bogdanov, D. V."https://www.zbmath.org/authors/?q=ai:bogdanov.d-v.1"Kytmanov, A. A."https://www.zbmath.org/authors/?q=ai:kytmanov.a-a"Sadykov, T. M."https://www.zbmath.org/authors/?q=ai:sadykov.t-mSummary: We present algorithms for computation and visualization of polynomial amoebas, their contours, compactified amoebas and sections of three-dimensional amoebas by two-dimensional planes. We also provide a method and an algorithm for the computation of polynomials whose amoebas exhibit the most complicated topology among all polynomials with a fixed Newton polytope. The presented algorithms are implemented in computer algebra systems Matlab 8 and Mathematica 9.
For the entire collection see [Zbl 1346.68010].Computing all space curve solutions of polynomial systems by polyhedral methods.https://www.zbmath.org/1453.130842021-02-27T13:50:00+00:00"Bliss, Nathan"https://www.zbmath.org/authors/?q=ai:bliss.nathan"Verschelde, Jan"https://www.zbmath.org/authors/?q=ai:verschelde.janSummary: A polyhedral method to solve a system of polynomial equations exploits its sparse structure via the Newton polytopes of the polynomials. We propose a hybrid symbolic-numeric method to compute a Puiseux series expansion for every space curve that is a solution of a polynomial system. The focus of this paper concerns the difficult case when the leading powers of the Puiseux series of the space curve are contained in the relative interior of a higher dimensional cone of the tropical prevariety. We show that this difficult case does not occur for polynomials with generic coefficients. To resolve this case, we propose to apply polyhedral end games to recover tropisms hidden in the tropical prevariety.
For the entire collection see [Zbl 1346.68010].An MCMC model search algorithm for regression problems.https://www.zbmath.org/1453.621772021-02-27T13:50:00+00:00"Petralias, Athanassios"https://www.zbmath.org/authors/?q=ai:petralias.athanassios"Dellaportas, Petros"https://www.zbmath.org/authors/?q=ai:dellaportas.petrosSummary: In this paper, we improve upon the Carlin and Chib Markov chain Monte Carlo algorithm that searches in model and parameter spaces. Our proposed algorithm attempts non-uniformly chosen `local' moves in the model space and avoids some pitfalls of other existing algorithms. In a series of examples with linear and logistic regression, we report evidence that our proposed algorithm performs better than the existing algorithms.Symbolic manipulation of flows of nonlinear evolution equations, with application in the analysis of split-step time integrators.https://www.zbmath.org/1453.651162021-02-27T13:50:00+00:00"Auzinger, Winfried"https://www.zbmath.org/authors/?q=ai:auzinger.winfried"Hofstätter, Harald"https://www.zbmath.org/authors/?q=ai:hofstatter.harald"Koch, Othmar"https://www.zbmath.org/authors/?q=ai:koch.othmarSummary: We describe a package realized in the Julia programming language which performs symbolic manipulations applied to nonlinear evolution equations, their flows, and commutators of such objects. This tool was employed to perform contrived computations arising in the analysis of the local error of operator splitting methods. It enabled the proof of the convergence of the basic method and of the asymptotical correctness of a defect-based error estimator. The performance of our package is illustrated on several examples.
For the entire collection see [Zbl 1346.68010].Setup of order conditions for splitting methods.https://www.zbmath.org/1453.651152021-02-27T13:50:00+00:00"Auzinger, Winfried"https://www.zbmath.org/authors/?q=ai:auzinger.winfried"Herfort, Wolfgang"https://www.zbmath.org/authors/?q=ai:herfort.wolfgang-n"Hofstätter, Harald"https://www.zbmath.org/authors/?q=ai:hofstatter.harald"Koch, Othmar"https://www.zbmath.org/authors/?q=ai:koch.othmarSummary: For operator splitting methods, an approach based on Taylor expansion and the particular structure of its leading term as an element of a free Lie algebra is used for the setup of a system of order conditions. Along with a brief review of the underlying theoretical background, we discuss the implementation of the resulting algorithm in computer algebra, in particular using Maple 18. A parallel version of such a code is described, and its performance on a computational node with 16 threads is documented.
For the entire collection see [Zbl 1346.68010].On the differential and full algebraic complexities of operator matrices transformations.https://www.zbmath.org/1453.340142021-02-27T13:50:00+00:00"Abramov, Sergei A."https://www.zbmath.org/authors/?q=ai:abramov.sergei-aSummary: We consider \(n\times n\)-matrices whose entries are scalar ordinary differential operators of order \(\le d\) over a constructive differential field \(K\). We show that to choose an algorithm to solve a problem related to such matrices it is reasonable to take into account the complexity measured as the number not only of arithmetic operations in \(K\) in the worst case but of all operations including differentiation. The algorithms that have the same complexity in terms of the number of arithmetic operations can though differ in the context of the full algebraic complexity that includes the necessary differentiations. Following this, we give a complexity analysis, first, of finding a superset of the set of singular points for solutions of a system of linear ordinary differential equations, and, second, of the unimodularity testing for an operator matrix and of constructing the inverse matrix if it exists.
For the entire collection see [Zbl 1346.68010].An accelerated version of Newton's method with convergence order \(\sqrt{3}+1\).https://www.zbmath.org/1453.651002021-02-27T13:50:00+00:00"McDougall, Trevor J."https://www.zbmath.org/authors/?q=ai:mcdougall.trevor-j"Wotherspoon, Simon J."https://www.zbmath.org/authors/?q=ai:wotherspoon.simon-j"Barker, Paul M."https://www.zbmath.org/authors/?q=ai:barker.paul-mSummary: A root-finding method is developed that, like Newton's Method, evaluates both the function and its first derivative once per iteration, but the new method converges at the rate \(\sqrt{ 3} + 1\), and moreover, it's asymptotic error constant is proportional to the function's fourth order derivative. By contrast, Newton's Method converges quadratically with the asymptotic error constant being proportional to the function's second order derivative. Each iteration (except the first) of our Accelerated Newton's Method (ANM) uses the values of both the function and its first derivative at the previous iteration in order to estimate the function's second derivative. For the initial iteration we develop and recommend the use of a modified version of Jarratt's Method; a method that calculates the derivative of the function twice in each iteration. Like Jarratt's Method, our modification of it converges with the fourth power of the initial error, but our asymptotic error constant depends primarily on the product of the function's second and third order derivatives rather than depending separately on the value of the second derivative. The efficient performance of our Accelerated Newton's Method (ANM) is illustrated using nine test functions and a range of initial values for each test function. These tests indicate that our Accelerated Newton's Method requires on average 30\% fewer function and derivative evaluations than the straightforward Newton's Method to achieve the same accuracy; noting again that the function and its derivative are evaluated once per iteration, exactly as in Newton's Method. Moreover, we find that our Accelerated Newton's Method is more robust than Newton's Method in that it converges to the root over a wider range of initial conditions.Global convergence via descent modified three-term conjugate gradient projection algorithm with applications to signal recovery.https://www.zbmath.org/1453.651202021-02-27T13:50:00+00:00"Abubakar, Auwal Bala"https://www.zbmath.org/authors/?q=ai:bala-abubakar.auwal"Kumam, Poom"https://www.zbmath.org/authors/?q=ai:kumam.poom"Awwal, Aliyu Muhammed"https://www.zbmath.org/authors/?q=ai:awwal.aliyu-muhammedSummary: In this article, we propose a three-term conjugate gradient projection algorithm for solving constrained monotone nonlinear equations. The global convergence of the algorithm was established under suitable assumptions. Numerical examples presented indicate that the algorithm has a very good performance in solving monotone nonlinear equations. Finally, the algorithm is applied to solve signal recovery problems.Quadrature formulae of Euler-Maclaurin type based on generalized Euler polynomials of level \(m\).https://www.zbmath.org/1453.650552021-02-27T13:50:00+00:00"Quintana, Yamilet"https://www.zbmath.org/authors/?q=ai:quintana.yamilet"Urieles, Alejandro"https://www.zbmath.org/authors/?q=ai:urieles.alejandroSummary: This article deals with some properties -- which are, to the best of our knowledge, new -- of the generalized Euler polynomials of level \(m\). These properties include a new recurrence relation satisfied by these polynomials and quadrature formulae of Euler-Maclaurin type based on them. Numerical examples are also given.Dispersion-dissipation analysis for advection problems with nonconstant coefficients: applications to discontinuous Galerkin formulations.https://www.zbmath.org/1453.653372021-02-27T13:50:00+00:00"Manzanero, Juan"https://www.zbmath.org/authors/?q=ai:manzanero.juan"Rubio, Gonzalo"https://www.zbmath.org/authors/?q=ai:rubio.gonzalo"Ferrer, Esteban"https://www.zbmath.org/authors/?q=ai:ferrer.esteban"Valero, Eusebi"https://www.zbmath.org/authors/?q=ai:valero.eusebiOn the modified methods for irreducible linear systems with L-matrices.https://www.zbmath.org/1453.650692021-02-27T13:50:00+00:00"Edalatpanah, Seyyed Ahmad"https://www.zbmath.org/authors/?q=ai:edalatpanah.seyyed-ahmadSummary: \textit{J. P. Milaszewicz} [Linear Algebra Appl. 93, 161--170 (1987; Zbl 0628.65022)] presented new preconditioner for linear system in order to improve the convergence rates of Jacobi and Gauss-Seidel iterative methods. \textit{Y.-T. Li} et al. [Appl. Math. Comput. 186, No. 1, 379--388 (2007; Zbl 1121.65032)] applied this preconditioner and provided convergence theorem for modified AOR method. \textit{J. H. Yun} and \textit{S. W. Kim} [ibid. 201, No. 1--2, 56--64 (2008; Zbl 1155.65324)] pointed out some errors in Li et al.'s [loc. cit.] theorem and provided some correct results for convergence of the preconditioned AOR method. In this paper, we analyze their convergence properly and propose a new theorem for irreducible modified AOR method. In particular, based on directed graph, we prove that the convergence theorem of Li et al. [loc. cit.] is true, without any additional assumptions.Constructive theory of functions. Proceedings of the 13th international conference, Sozopol, Bulgaria, June 2--8, 2019. Dedicated to the memory of Blagovest Sendov.https://www.zbmath.org/1453.410012021-02-27T13:50:00+00:00"Draganov, Borislav (ed.)"https://www.zbmath.org/authors/?q=ai:draganov.borislav-r"Ivanov, Kamen (ed.)"https://www.zbmath.org/authors/?q=ai:ivanov.kamen-g"Nikolov, Geno (ed.)"https://www.zbmath.org/authors/?q=ai:nikolov.geno-p"Uluchev, Rumen (ed.)"https://www.zbmath.org/authors/?q=ai:uluchev.rumen-kThe articles of this volume will be reviewed individually. For the preceding conference see [Zbl 1412.41002].Some notes on summation by parts time integration methods.https://www.zbmath.org/1453.651612021-02-27T13:50:00+00:00"Ranocha, Hendrik"https://www.zbmath.org/authors/?q=ai:ranocha.hendrikSummary: Some properties of numerical time integration methods using summation by parts (SBP) operators and simultaneous approximation terms are studied. These schemes can be interpreted as implicit Runge-Kutta methods with desirable stability properties such as \(A\)-,\(B\)-,\(L\)-, and algebraic stability [\textit{J. Nordström} and \textit{T. Lundquist}, J. Comput. Phys. 251, 487--499 (2013; Zbl 1349.65399); \textit{T. Lundquist} and \textit{J. Nordström}, J. Comput. Phys. 270, 86--104 (2014; Zbl 1349.65550); \textit{P. D. Boom} and \textit{D. W. Zingg}, SIAM J. Sci. Comput. 37, No. 6, A2682--A2709 (2015; Zbl 1359.65127); \textit{A. A. Ruggiu} and \textit{J. Nordström}, J. Comput. Phys. 360, 192--201 (2018; Zbl 1395.65107)]. Here, insights into the necessity of certain assumptions, relations to known Runge-Kutta methods, and stability properties are provided by new proofs and counterexamples. In particular, it is proved that a) a technical assumption is necessary since it is not