×

zbMATH — the first resource for mathematics

Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes. (English) Zbl 1378.76044
Summary: We present a new family of high order accurate fully discrete one-step discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes, like molecular viscosity or heat conduction. High order piecewise polynomials of degree \(N\) are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach, making use of an element-local space-time Galerkin finite element predictor. A novel nodal solver algorithm based on the HLL flux is derived to compute the velocity for each nodal degree of freedom that describes the current mesh geometry. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Each technique generates a corresponding number of geometrical degrees of freedom needed to describe the current mesh configuration and which must be considered by the nodal solver for determining the grid velocity. The connection of the old mesh configuration at time \(t^n\) with the new one at time \(t^{n + 1}\) provides the 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 numerical method belongs to the category of so-called direct arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry (including a possible rezoning step) directly during the computation of the numerical fluxes. We emphasize that our method is a moving mesh method, as opposed to total Lagrangian formulations that are based on a fixed computational grid and which instead evolve the mapping of the reference configuration to the current one. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method recently developed in [the second author and R. Loubère, ibid. 319, 163–199 (2016; Zbl 1349.65447)] for fixed unstructured grids. In this approach, the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria, such as the positivity of pressure and density, the absence of floating point errors (NaN) and the satisfaction of a relaxed discrete maximum principle (DMP) in the sense of polynomials. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed at the aid of a more robust second order TVD finite volume scheme. To preserve the subcell resolution capability of the original DG scheme, the FV limiter is run on a sub-grid that is \(2 N + 1\) times finer compared to the mesh of the original unlimited DG scheme. The new subcell averages are then gathered back into a high order DG polynomial by a usual conservative finite volume reconstruction operator. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated in order to check the accuracy and the robustness of the proposed numerical method in the context of the Euler and Navier-Stokes equations for compressible gas dynamics, considering both inviscid and viscous fluids. Finally, an application inspired by inertial confinement fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76W05 Magnetohydrodynamics and electrohydrodynamics
Software:
HE-E1GODF; HLLE; ReALE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Balsara, D. S.; Rumpf, T.; Dumbser, M.; Munz, C. D., Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics, J. Comput. Phys., 228, 2480-2516, (2009) · Zbl 1275.76169
[2] Barth, T. J.; Jespersen, D. C., The design and application of upwind schemes on unstructured meshes, 1-12, (1989), AIAA Paper 89-0366
[3] Bassi, F.; Botti, L.; Colombo, A.; Di Pietro, D. A.; Tesini, P., On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations, J. Comput. Phys., 231, 45-65, (2012) · Zbl 1457.65178
[4] Bassi, F.; Botti, L.; Colombo, A.; Rebay, S., Agglomeration based discontinuous Galerkin discretization of the Euler and Navier-Stokes equations, Comput. Fluids, 61, 77-85, (2012) · Zbl 1365.76109
[5] Bassi, F.; Crivellini, A.; Di Pietro, D. A.; Rebay, S., An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows, Comput. Fluids, 36, 1529-1546, (2007) · Zbl 1194.76102
[6] Bassi, F.; Franchina, N.; Ghidoni, A.; Rebay, S., Spectral p-multigrid discontinuous Galerkin solution of the Navier-Stokes equations, Int. J. Numer. Methods Fluids, 67, 1540-1558, (2011) · Zbl 1426.76497
[7] Becker, R., Stosswelle und detonation, Physik, 8, 321, (1923)
[8] Benson, D. J., Momentum advection on a staggered mesh, J. Comput. Phys., 100, 1, 143-162, (1992) · Zbl 0758.76038
[9] Berndt, M.; Breil, J.; Galera, S.; Kucharik, M.; Maire, P. H.; Shashkov, M., Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods, J. Comput. Phys., 230, 6664-6687, (2011) · Zbl 1408.65077
[10] Bochev, P.; Ridzal, D.; Shashkov, M. J., Fast optimization-based conservative remap of scalar fields through aggregate mass transfer, J. Comput. Phys., 246, 37-57, (2013) · Zbl 1349.65054
[11] Boscheri, W., An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics, Int. J. Numer. Methods Fluids, 134-135, 111-129, (2016)
[12] Boscheri, W., High order direct arbitrary-Lagrangian-Eulerian (ALE) finite volume schemes for hyperbolic systems on unstructured meshes, Arch. Comput. Methods Eng., 1-51, (2016)
[13] Boscheri, W.; Balsara, D. S.; Dumbser, M., Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers, J. Comput. Phys., 267, 112-138, (2014) · Zbl 1349.76309
[14] Boscheri, W.; Dumbser, M., Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes, Commun. Comput. Phys., 14, 1174-1206, (2013) · Zbl 1388.65075
[15] Boscheri, W.; Dumbser, M., A direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D, J. Comput. Phys., 275, 484-523, (2014) · Zbl 1349.76310
[16] Boscheri, W.; Dumbser, M., An efficient quadrature-free formulation for high order arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on unstructured meshes, J. Sci. Comput., 66, 240-274, (2016) · Zbl 1338.65219
[17] Boscheri, W.; Dumbser, M., High order accurate direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on moving curvilinear unstructured meshes, Comput. Fluids, 136, 48-66, (2016) · Zbl 1390.76398
[18] Boscheri, W.; Dumbser, M.; Balsara, D. S., High order Lagrangian ADER-WENO schemes on unstructured meshes - application of several node solvers to hydrodynamics and magnetohydrodynamics, Int. J. Numer. Methods Fluids, 76, 737-778, (2014)
[19] Boscheri, W.; Dumbser, M.; Loubère, R., Cell centered direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity, Comput. Fluids, 134-135, 111-129, (2016) · Zbl 1390.76399
[20] Boscheri, W.; Dumbser, M.; Zanotti, O., High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes, J. Comput. Phys., 291, 120-150, (2014) · Zbl 1349.76311
[21] Boscheri, W.; Loubère, R., High order accurate direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for non-conservative hyperbolic systems with stiff source terms, Commun. Comput. Phys., 21, 271-312, (2017) · Zbl 1388.65076
[22] Boscheri, W.; Loubère, R.; Dumbser, M., Direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws, J. Comput. Phys., 292, 56-87, (2015) · Zbl 1349.76312
[23] Breil, J.; Harribey, T.; Maire, P. H.; Shashkov, M. J., A multi-material reale method with MOF interface reconstruction, Comput. Fluids, 83, 115-125, (2013) · Zbl 1290.76094
[24] Caramana, E. J.; Burton, D. E.; Shashkov, M. J.; Whalen, P. P., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys., 146, 227-262, (1998) · Zbl 0931.76080
[25] Carré, G.; Del Pino, S.; Després, B.; Labourasse, E., A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, J. Comput. Phys., 228, 5160-5183, (2009) · Zbl 1168.76029
[26] Casoni, E.; Peraire, J.; Huerta, A., One-dimensional shock-capturing for high-order discontinuous Galerkin methods, Int. J. Numer. Methods Fluids, 71, 6, 737-755, (2013)
[27] Castro, C. C.; Toro, E. F., Solvers for the high-order Riemann problem for hyperbolic balance laws, J. Comput. Phys., 227, 2481-2513, (2008) · Zbl 1148.65066
[28] Cheng, J.; Shu, C. W., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. Comput. Phys., 227, 1567-1596, (2007) · Zbl 1126.76035
[29] Cheng, J.; Shu, C. W., A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry, J. Comput. Phys., 229, 7191-7206, (2010) · Zbl 1425.35142
[30] Clain, S.; Diot, S.; Loubère, R., A high-order finite volume method for systems of conservation laws - multi-dimensional optimal order detection (MOOD), J. Comput. Phys., 230, 10, 4028-4050, (2011) · Zbl 1218.65091
[31] Claisse, A.; Després, B.; Labourasse, E.; Ledoux, F., A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes, J. Comput. Phys., 231, 4324-4354, (2012) · Zbl 1426.76350
[32] Cockburn, B.; Hou, S.; Shu, C. W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54, 545-581, (1990) · Zbl 0695.65066
[33] Cockburn, B.; Karniadakis, G. E.; Shu, C. W., Discontinuous Galerkin methods, Lecture Notes in Computational Science and Engineering, (2000), Springer
[34] Cockburn, B.; Lin, S. Y.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, J. Comput. Phys., 84, 90-113, (1989) · Zbl 0677.65093
[35] Cockburn, B.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52, 411-435, (1989) · Zbl 0662.65083
[36] Cockburn, B.; Shu, C. W., The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, Math. Model. Numer. Anal., 25, 337-361, (1991) · Zbl 0732.65094
[37] Cockburn, B.; Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059
[38] Crivellini, A.; Bassi, F., An implicit matrix-free discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations, Comput. Fluids, 50, 81-93, (2011) · Zbl 1271.76164
[39] Després, B.; Mazeran, C., Symmetrization of Lagrangian gas dynamic in dimension two and multidimensional solvers, C. R., Méc., 331, 475-480, (2003) · Zbl 1293.76089
[40] Després, B.; Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. Ration. Mech. Anal., 178, 327-372, (2005) · Zbl 1096.76046
[41] Diot, S.; Clain, S.; Loubère, R., Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials, Comput. Fluids, 64, 43-63, (2012) · Zbl 1365.76149
[42] Diot, S.; Loubère, R.; Clain, S., The MOOD method in the three-dimensional case: very-high-order finite volume method for hyperbolic systems, Int. J. Numer. Methods Fluids, 73, 362-392, (2013)
[43] Dobrev, V.; Kolev, T.; Rieben, R., High-order curvilinear finite element methods for Lagrangian hydrodynamics, SIAM J. Sci. Comput., 34, 5, B606-B641, (2012) · Zbl 1255.76076
[44] Dobrev, V. A.; Ellis, T. E.; Kolev, T. V.; Rieben, R. N., High-order curvilinear finite elements for axisymmetric Lagrangian hydrodynamics, Comput. Fluids, 83, 58-69, (2013) · Zbl 1290.76061
[45] Dobrev, V. A.; Ellis, T. E.; Kolev, Tz. V.; Rieben, R. N., Curvilinear finite elements for Lagrangian hydrodynamics, Int. J. Numer. Methods Fluids, 65, 1295-1310, (2011) · Zbl 1255.76075
[46] Dolejsi, V., Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows, Commun. Comput. Phys., 4, 231-274, (2008) · Zbl 1364.76085
[47] Dolejsi, V.; Feistauer, M., A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow, J. Comput. Phys., 198, 727-746, (2004) · Zbl 1116.76386
[48] Dolejsi, V.; Feistauer, M.; Hozman, J., Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes, Comput. Methods Appl. Mech. Eng., 196, 2813-2827, (2007) · Zbl 1121.76033
[49] Dubiner, M., Spectral methods on triangles and other domains, J. Sci. Comput., 6, 345-390, (1991) · Zbl 0742.76059
[50] Dukovicz, J. K.; Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J. Comput. Phys., 99, 115-134, (1992) · Zbl 0743.76058
[51] Dumbser, M., Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations, Comput. Fluids, 39, 60-76, (2010) · Zbl 1242.76161
[52] Dumbser, M., Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws, Comput. Methods Appl. Mech. Eng., 280, 57-83, (2014) · Zbl 1423.76296
[53] Dumbser, M.; Balsara, D.; Toro, E. F.; Munz, C. D., A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes, J. Comput. Phys., 227, 8209-8253, (2008) · Zbl 1147.65075
[54] Dumbser, M.; Balsare, D. S., High-order unstructured one-step P N P M schemes for the viscous and resistive MHD equations, Comput. Model. Eng. Sci., 54, 301-332, (2009) · Zbl 1231.76345
[55] Dumbser, M.; Boscheri, W., High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows, Comput. Fluids, 86, 405-432, (2013) · Zbl 1290.76081
[56] Dumbser, M.; Castro, M.; Parés, C.; Toro, E. F., ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows, Comput. Fluids, 38, 1731-1748, (2009) · Zbl 1177.76222
[57] Dumbser, M.; Enaux, C.; Toro, E. F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227, 3971-4001, (2008) · Zbl 1142.65070
[58] Dumbser, M.; Hidalgo, A.; Castro, M.; Parés, C.; Toro, E. F., FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems, Comput. Methods Appl. Mech. Eng., 199, 625-647, (2010) · Zbl 1227.76043
[59] Dumbser, M.; Kaeser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221, 693-723, (2007) · Zbl 1110.65077
[60] Dumbser, M.; Kaeser, M.; Titarev, V. A.; Toro, E. F., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys., 226, 204-243, (2007) · Zbl 1124.65074
[61] Dumbser, M.; Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221, 693-723, (2007) · Zbl 1110.65077
[62] Dumbser, M.; Loubère, R., A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes, J. Comput. Phys., 319, 163-199, (2016) · Zbl 1349.65447
[63] Dumbser, M.; Peshkov, I.; Romenski, E.; Zanotti, O., High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids, J. Comput. Phys., 314, 824-862, (2016) · Zbl 1349.76324
[64] Dumbser, M.; Uuriintsetseg, A.; Zanotti, O., On arbitrary-Lagrangian-Eulerian one-step WENO schemes for stiff hyperbolic balance laws, Commun. Comput. Phys., 14, 301-327, (2013) · Zbl 1373.76126
[65] Dumbser, M.; Zanotti, O.; Loubère, R.; Diot, S., A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws, J. Comput. Phys., 278, 47-75, (2014) · Zbl 1349.65448
[66] Francois, M. M.; Shashkov, M. J.; Masser, T. O.; Dendy, E. D., A comparative study of multimaterial Lagrangian and Eulerian methods with pressure relaxation, Comput. Fluids, 83, 126-136, (2013) · Zbl 1290.76133
[67] Friess, M. B.; Breil, J.; Maire, P. H.; Shashkov, M., A multi-material CCALE-MOF approach in cylindrical geometry, Commun. Comput. Phys., 15, 330-364, (2014) · Zbl 1388.65078
[68] Vilar, F., Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics, Comput. Fluids, 64, 64-73, (2012) · Zbl 1365.76129
[69] Vilar, F.; Maire, P. H.; Abgrall, R., Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics, Comput. Fluids, 46, 1, 498-604, (2010) · Zbl 1433.76093
[70] Vilar, F.; Maire, P. H.; Abgrall, R., A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids, J. Comput. Phys., 276, 188-234, (2014) · Zbl 1349.76278
[71] Galera, S.; Maire, P. H.; Breil, J., A two-dimensional unstructured cell-centered multi-material ale scheme using vof interface reconstruction, J. Comput. Phys., 229, 5755-5787, (2010) · Zbl 1346.76105
[72] Gassner, G.; Lörcher, F.; Munz, C. D., A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes, J. Comput. Phys., 224, 1049-1063, (2007) · Zbl 1123.76040
[73] Giraldo, F. X.; Restelli, M., High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model, Int. J. Numer. Methods Fluids, 63, 1077-1102, (2010) · Zbl 1267.76010
[74] Godunov, S. K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Math. USSR, 47, 271-306, (1959) · Zbl 0171.46204
[75] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61, (1983) · Zbl 0565.65051
[76] Hidalgo, A.; Dumbser, M., ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations, J. Sci. Comput., 48, 173-189, (2011) · Zbl 1221.65231
[77] Hirt, C. W.; Amsden, A. A.; Cook, J. L., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys., 14, 227-253, (1974) · Zbl 0292.76018
[78] Hu, C.; Shu, C. W., A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys., 150, 561-594, (1999) · Zbl 0937.76051
[79] Hu, C.; Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 97-127, (1999) · Zbl 0926.65090
[80] Huerta, A.; Casoni, E.; Peraire, J., A simple shock-capturing technique for high-order discontinuous Galerkin methods, Int. J. Numer. Methods Fluids, 69, 10, 1614-1632, (2012) · Zbl 1253.76058
[81] Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 202-228, (1996) · Zbl 0877.65065
[82] Karniadakis, G. E.; Sherwin, S. J., Spectral/hp element methods in CFD, (1999), Oxford University Press · Zbl 0924.76078
[83] Kidder, R. E., Laser-driven compression of hollow shells: power requirements and stability limitations, Nucl. Fusion, 1, 3-14, (1976)
[84] Knupp, P. M., Achieving finite element mesh quality via optimization of the jacobian matrix norm and associated quantities. part II - A framework for volume mesh optimization and the condition number of the Jacobian matrix, Int. J. Numer. Methods Eng., 48, 1165-1185, (2000) · Zbl 0990.74069
[85] Kucharik, M.; Breil, J.; Galera, S.; Maire, P. H.; Berndt, M.; Shashkov, M. J., Hybrid remap for multi-material ALE, Comput. Fluids, 46, 293-297, (2011) · Zbl 1433.76133
[86] Kucharik, M.; Shashkov, M. J., One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian-Eulerian methods, J. Comput. Phys., 231, 2851-2864, (2012) · Zbl 1323.74108
[87] Li, Z.; Yu, X.; Jia, Z., The cell-centered discontinuous Galerkin method for Lagrangian compressible Euler equations in two dimensions, Comput. Fluids, 96, 152-164, (2014) · Zbl 1391.76347
[88] Liska, R.; Shashkov, M.; Váchal, P.; Wendroff, B., Synchronized flux corrected remapping for ALE methods, Comput. Fluids, 46, 312-317, (2011) · Zbl 1433.76135
[89] Liu, W.; Cheng, J.; Shu, C. W., High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations, J. Comput. Phys., 228, 8872-8891, (2009) · Zbl 1287.76181
[90] Loubère, R.; Maire, P. H.; Váchal, P., Staggered Lagrangian hydrodynamics based on cell-centered Riemann solver, Commun. Comput. Phys., 10, 4, 940-978, (2010) · Zbl 1373.76138
[91] Loubère, R.; Maire, P. H.; Váchal, P., A second-order compatible staggered Lagrangian hydrodynamics scheme using a cell-centered multidimensional approximate Riemann solver, Proc. Comput. Sci., 1, 1931-1939, (2010) · Zbl 1432.76206
[92] Loubère, R.; Maire, P. H.; Váchal, P., 3D staggered Lagrangian hydrodynamics scheme with cell-centered Riemann solver-based artificial viscosity, Int. J. Numer. Methods Fluids, 72, 22-42, (2013)
[93] Maire, P. H., A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry, J. Comput. Phys., 228, 18, 6882-6915, (2009) · Zbl 1261.76021
[94] Maire, P. H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured mesh, J. Comput. Phys., 228, 7, 2391-2425, (2009) · Zbl 1156.76434
[95] Maire, P. H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for compressible flow problems, SIAM J. Sci. Comput., 29, 4, 1781-1824, (2007) · Zbl 1251.76028
[96] Maire, P. H., A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Comput. Fluids, 46, 1, 341-347, (2011) · Zbl 1433.76137
[97] Maire, P. H., A unified sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Int. J. Numer. Methods Fluids, 65, 1281-1294, (2011) · Zbl 1429.76089
[98] Maire, P. H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput., 29, 1781-1824, (2007) · Zbl 1251.76028
[99] Maire, P. H.; Nkonga, B., Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics, J. Comput. Phys., 228, 799-821, (2009) · Zbl 1156.76039
[100] Meister, A.; Ortleb, S., A positivity preserving and well-balanced DG scheme using finite volume subcells in almost dry regions, Appl. Math. Comput., 272, 259-273, (2016) · Zbl 1410.76250
[101] Millington, R. C.; Toro, E. F.; Nejad, L. A.M., Arbitrary high order methods for conservation laws I: the one dimensional scalar case, (June 1999), Manchester Metropolitan University, Department of Computing and Mathematics, PhD thesis
[102] Munz, C. D., On Godunov-type schemes for Lagrangian gas dynamics, SIAM J. Numer. Anal., 31, 17-42, (1994) · Zbl 0796.76057
[103] Von Neumann, J.; Richtmyer, R. D., A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21, 232-237, (1950) · Zbl 0037.12002
[104] Nigro, A.; Renda, S.; De Bartolo, C.; Hartmann, R.; Bassi, F., A high-order accurate discontinuous Galerkin finite element method for laminar low Mach number flows, Int. J. Numer. Methods Fluids, 72, 43-68, (2013)
[105] López Ortega, A.; Scovazzi, G., A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian-Eulerian computations with nodal finite elements, J. Comput. Phys., 230, 6709-6741, (2011) · Zbl 1284.76255
[106] Peery, J. S.; Carroll, D. E., Multi-material ale methods in unstructured grids, Comput. Methods Appl. Mech. Eng., 187, 591-619, (2000) · Zbl 0980.74068
[107] Peshkov, I.; Romenski, E., A hyperbolic model for viscous Newtonian flows, Contin. Mech. Thermodyn., 28, 85-104, (2016) · Zbl 1348.76046
[108] Qiu, J.; Dumbser, M.; Shu, C. W., The discontinuous Galerkin method with Lax-Wendroff type time discretizations, Comput. Methods Appl. Mech. Eng., 194, 4528-4543, (2005) · Zbl 1093.76038
[109] Reed, W. H.; Hill, T. R., Triangular mesh methods for neutron transport equation, (1973), Los Alamos Scientific Laboratory, Technical Report LA-UR-73-479
[110] Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys. USSR, 1, 267-279, (1961)
[111] Sambasivan, S. K.; Shashkov, M. J.; Burton, D. E., A finite volume cell-centered Lagrangian hydrodynamics approach for solids in general unstructured grids, Int. J. Numer. Methods Fluids, 72, 770-810, (2013)
[112] Sambasivan, S. K.; Shashkov, M. J.; Burton, D. E., Exploration of new limiter schemes for stress tensors in Lagrangian and ALE hydrocodes, Comput. Fluids, 83, 98-114, (2013) · Zbl 1290.76107
[113] Scovazzi, G., Lagrangian shock hydrodynamics on tetrahedral meshes: a stable and accurate variational multiscale approach, J. Comput. Phys., 231, 8029-8069, (2012)
[114] Smith, R. W., AUSM(ALE): a geometrically conservative arbitrary Lagrangian-Eulerian flux splitting scheme, J. Comput. Phys., 150, 268-286, (1999) · Zbl 0936.76046
[115] Sonntag, M.; Munz, C. D., Shock capturing for discontinuous Galerkin methods using finite volume subcells, (Fuhrmann, J.; Ohlberger, M.; Rohde, C., Finite Volumes for Complex Applications VII, (2014), Springer), 945-953 · Zbl 1426.76429
[116] Sonntag, M.; Munz, C. D., Efficient parallelization of a shock capturing for discontinuous Galerkin methods using finite volume sub-cells, J. Sci. Comput., 70, 1262-1289, (2017) · Zbl 1366.65089
[117] Springel, V., E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh, Mon. Not. R. Astron. Soc., 401, 791-851, (2010)
[118] Stroud, A. H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Inc. Englewood Cliffs, New Jersey · Zbl 0379.65013
[119] Tavelli, M.; Dumbser, M., A staggered arbitrary high order semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier-Stokes equations, Appl. Math. Comput., 248, 70-92, (2014) · Zbl 1338.76068
[120] Tavelli, M.; Dumbser, M., A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes, J. Comput. Phys., 319, 294-323, (2016) · Zbl 1349.76271
[121] Titarev, V. A.; Toro, E. F., ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 1-4, 609-618, (December 2002)
[122] Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J. Comput. Phys., 204, 715-736, (2005) · Zbl 1060.65641
[123] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer · Zbl 0923.76004
[124] Toro, E. F., Anomalies of conservative methods: analysis, numerical evidence and possible cures, Int. J. Comput. Fluid Dyn., 11, 128-143, (2002)
[125] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: A practical introduction, (2009), Springer · Zbl 1227.76006
[126] van Leer, B., Toward the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136, (1979) · Zbl 1364.65223
[127] Vilar, F.; Shu, C. W.; Maire, P. H., Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: from first-order to high-orders. part II: the two-dimensional case, J. Comput. Phys., 312, 416-442, (2016) · Zbl 1351.76128
[128] Yanilkin, Y. V.; Goncharov, E. A.; Kolobyanin, V. Y.; Sadchikov, V. V.; Kamm, J. R.; Shashkov, M. J.; Rider, W. J., Multi-material pressure relaxation methods for Lagrangian hydrodynamics, Comput. Fluids, 83, 137-143, (2013) · Zbl 1290.76138
[129] Zanotti, O.; Fambri, F.; Dumbser, M.; Hidalgo, A., Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting, Comput. Fluids, 118, 204-224, (2015) · Zbl 1390.76381
[130] Zhang, Y. T.; Shu, C. W., Third order WENO scheme on three dimensional tetrahedral meshes, Commun. Comput. Phys., 5, 836-848, (2009) · Zbl 1364.65177
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.