×

zbMATH — the first resource for mathematics

Central weighted ENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes. (English) Zbl 1377.65115

MSC:
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
76N15 Gas dynamics, general
76M12 Finite volume methods applied to problems in fluid mechanics
65Y05 Parallel numerical computation
Software:
LIBXSMM
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. Abgrall, On essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation, J. Comput. Phys., 144 (1994), pp. 45–58. · Zbl 0822.65062
[2] T. Aboiyar, E. Georgoulis, and A. Iske, Adaptive ADER methods using kernel-based polyharmonic spline WENO reconstruction, SIAM J. Sci. Comput., 32 (2010), pp. 3251–3277. · Zbl 1221.65236
[3] D. Balsara, S. Garain, and C. Shu, An efficient class of WENO schemes with adaptive order, J. Comput. Phys., 326 (2016), pp. 780–804. · Zbl 1422.65146
[4] D. Balsara and C. Shu, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160 (2000), pp. 405–452. · Zbl 0961.65078
[5] T. Barth and P. Frederickson, Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA Paper 90-0013, 1990.
[6] T. Barth and D. Jespersen, The Design and Application of Upwind Schemes on Unstructured Meshes, AIAA Paper 89-0366, 1989.
[7] W. Boscheri and M. Dumbser, 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 (2014), pp. 484–523. · Zbl 1349.76310
[8] W. Boscheri and M. Dumbser, An efficient quadrature-free formulation for high order arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on unstructured meshes, J. Sci. Comput., 66 (2016), pp. 240–274. · Zbl 1338.65219
[9] W. Boscheri, M. Dumbser, and D. Balsara, High order Lagrangian ADER-WENO schemes on unstructured meshes—Application of several node solvers to hydrodynamics and magnetohydrodynamics, Internat J. Numer. Methods Fluids, 76 (2014), pp. 737–778. · Zbl 1349.76309
[10] W. Boscheri, M. Dumbser, and R. Loubère, Cell centered direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity, Comput. & Fluids, 134-135 (2016), pp. 111–129. · Zbl 1390.76399
[11] P. Buchmüller, J. Dreher, and C. Helzel, Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement, Appl. Math. Comput., 272 (2016), pp. 460–478.
[12] P. Buchmüller and C. Helzel, Improved accuracy of high-order WENO finite volume methods on Cartesian grids, J. Sci. Comput., 61 (2014), pp. 343–368.
[13] G. Capdeville, A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes, J. Comput. Phys., 227 (2008), pp. 2977–3014. · Zbl 1135.65359
[14] E. Caramana, C. Rousculp, and D. Burton, A compatible, energy and symmetry preserving Lagrangian hydrodynamics algorithm in three-dimensional cartesian geometry, J. Comput. Phys., 157 (2000), pp. 89–119. · Zbl 0961.76049
[15] G. Carré, S. D. Pino, B. Després, and E. Labourasse, A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, J. Comput. Phys., 228 (2009), pp. 5160–5183. · Zbl 1168.76029
[16] C. C. Castro and E. F. Toro, Solvers for the high-order riemann problem for hyperbolic balance laws, J. Comput. Phys., 227 (2008), pp. 2481–2513. · Zbl 1148.65066
[17] M. Castro, J. Gallardo, and C. Parés, High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems, Math. Comp., 75 (2006), pp. 1103–1134. · Zbl 1096.65082
[18] J. Cheng and C. Shu, A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. Comput. Phys., 227 (2007), pp. 1567–1596. · Zbl 1126.76035
[19] J. Cheng, E. Toro, S. Jiang, and W. Tang, A sub-cell WENO reconstruction method for spatial derivatives in the ADER scheme, J. Comput. Phys., 251 (2013), pp. 53–80. · Zbl 1349.65279
[20] I. Cravero, G. Puppo, M. Semplice, and G. Visconti, CWENO: Uniformly Accurate Reconstructions for Balance Laws, Math. Comp., to appear; also available from . · Zbl 1412.65102
[21] I. Cravero and M. Semplice, On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes, J. Sci. Comput., 67 (2016), pp. 1219–1246. · Zbl 1343.65116
[22] M. Dubiner, Spectral methods on triangles and other domains, J. Sci. Comput., 6 (1991), pp. 345–390. · Zbl 0742.76059
[23] J. Dukovicz and B. Meltz, Vorticity errors in multidimensional Lagrangian codes, J. Comput. Phys., 99 (1992), pp. 115–134. · Zbl 0743.76058
[24] M. Dumbser, D. Balsara, E. Toro, and C. Munz, A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes, J. Comput. Phys., 227 (2008), pp. 8209–8253. · Zbl 1147.65075
[25] M. Dumbser, C. Enaux, and E. Toro, Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227 (2008), pp. 3971–4001. · Zbl 1142.65070
[26] M. Dumbser and M. Käser, Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221 (2006), pp. 693–723. · Zbl 1110.65077
[27] M. Dumbser, M. Käser, V. Titarev, and E. Toro, Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys., 226 (2007), pp. 204–243. · Zbl 1124.65074
[28] M. Dumbser and R. Loubère, A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes, J. Comput. Phys., 319 (2016), pp. 163–199. · Zbl 1349.65447
[29] M. Dumbser, I. Peshkov, E. Romenski, and O. Zanotti, 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 (2016), pp. 824–862. · Zbl 1349.76324
[30] M. Dumbser and E. F. Toro, On universal Osher–type schemes for general nonlinear hyperbolic conservation laws, Commun. Comput. Phys., 10 (2011), pp. 635–671. · Zbl 1373.76125
[31] M. Dumbser, O. Zanotti, A. Hidalgo, and D. Balsara, ADER-WENO finite volume schemes with space-time adaptive mesh refinement, J. Comput. Phys., 248 (2013), pp. 257–286. · Zbl 1349.76325
[32] O. Friedrich, Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys., 144 (1998), pp. 194–212. · Zbl 1392.76048
[33] S. Godunov and E. Romenski, Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates, J. Appl. Mech. Tech. Phys., 13 (1972), pp. 868–885.
[34] S. Godunov and E. Romenski, Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media, Comput. Fluid Dynamics Rev., 95 (1995), pp. 19–31. · Zbl 0875.73025
[35] C. R. Goetz and A. Iske, Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws, Math. Comp., 85 (2016), pp. 35–62. · Zbl 1325.65128
[36] A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), pp. 231–303. · Zbl 0652.65067
[37] A. Heinecke, H. Pabst, and G. Henry, LIBXSMM: A High Performance Library for Small Matrix Multiplications, presented at SC’15: The International Conference for High Performance Computing, Networking, Storage and Analysis, Austin, TX, 2015.
[38] C. Hu and C. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150 (1999), pp. 97–127. · Zbl 0926.65090
[39] G. Jiang and C. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), pp. 202–228. · Zbl 0877.65065
[40] G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods in CFD, Oxford University Press, New York, 1999. · Zbl 0954.76001
[41] G. Karypis and V. Kumar, Multilevel k-way partitioning scheme for irregular graphs, J. Parallel Distrib. Comput., 48 (1998), pp. 96–129. · Zbl 0918.68073
[42] M. Käser and A. Iske, ADER schemes on adaptive triangular meshes for scalar conservation laws, J. Comput. Phys., 205 (2005), pp. 486–508. · Zbl 1072.65116
[43] P. Knupp, Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities., Internat. J. Numer. Methods Engrg., 48 (2000), pp. 1165–1185. · Zbl 0990.74069
[44] O. Kolb, On the full and global accuracy of a compact third order WENO scheme, SIAM J. Numer. Anal., 52 (2014), pp. 2335–2355. · Zbl 1408.65062
[45] D. Levy, G. Puppo, and G. Russo, Central WENO schemes for hyperbolic systems of conservation laws, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 547–571. · Zbl 0938.65110
[46] D. Levy, G. Puppo, and G. Russo, A third order central WENO scheme for 2D conservation laws, Appl. Numer. Math., 33 (2000), pp. 415–421. · Zbl 0965.65106
[47] D. Levy, G. Puppo, and G. Russo, A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws, SIAM J. Sci. Comput., 24 (2002), pp. 480–506. · Zbl 1014.65079
[48] W. Liu, J. Cheng, and C. Shu, High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations, J. Comput. Phys., 228 (2009), pp. 8872–8891. · Zbl 1287.76181
[49] R. Loubère, P. Maire, and P. Váchal, 3D staggered Lagrangian hydrodynamics scheme with cell-centered Riemann solver-based artificial viscosity, Internat. J. Numer. Methods. Fluids, 72 (2013), pp. 22–42.
[50] P. Maire, A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Comput. & Fluids, 46 (2011), pp. 341–347. · Zbl 1433.76137
[51] P. Maire, R. Abgrall, J. Breil, and J. Ovadia, A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput., 29 (2007), pp. 1781–1824. · Zbl 1251.76028
[52] C. Parés, Numerical methods for nonconservative hyperbolic systems: A theoretical framework, SIAM J. Numer. Anal., 44 (2006), pp. 300–321.
[53] I. Peshkov and E. Romenski, A hyperbolic model for viscous Newtonian flows, Contin. Mech. Thermodyn., 28 (2016), pp. 85–104. · Zbl 1348.76046
[54] G. Puppo and M. Semplice, Numerical entropy and adaptivity for finite volume schemes, Commun. Comput. Phys., 10 (2011), pp. 1132–1160. · Zbl 1373.76140
[55] G. Puppo and M. Semplice, Well-balanced high order 1D schemes on non-uniform grids and entropy residuals, J. Sci. Comput., 66 (2016), pp. 1052–1076. · Zbl 1371.65093
[56] J. Qiu and C. Shu, On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes, J. Comput. Phys., 183 (2002), pp. 187–209. · Zbl 1018.65106
[57] E. Romenski, Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics, Math. Comput. Modeling, 28(10) (1998), pp. 115–130. · Zbl 1076.74501
[58] L. Sedov, Similarity and Dimensional Methods in Mechanics, Academic Press, New York, 1959. · Zbl 0121.18504
[59] M. Semplice, A. Coco, and G. Russo, Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction, J. Sci. Comput., 66 (2016), pp. 692–724. · Zbl 1335.65077
[60] J. Shi, C. Hu, and C. Shu, A technique of treating negative weights in WENO schemes, J. Comput. Phys., 175 (2002), pp. 108–127. · Zbl 0992.65094
[61] C. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51 (2009), pp. 82–126. · Zbl 1160.65330
[62] C. Shu, High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments, J. Comput. Phys., 316 (2016), pp. 598–613. · Zbl 1349.65486
[63] V. Titarev and E. Toro, ADER: Arbitrary high order Godunov approach, J. Sci. Comput., 17 (2002), pp. 609–618. · Zbl 1024.76028
[64] V. Titarev and E. Toro, Finite-volume WENO schemes for three-dimensional conservation laws, J. Comput. Phys., 201 (2004), pp. 238–260. · Zbl 1059.65078
[65] V. Titarev, P. Tsoutsanis, and D. Drikakis, WENO schemes for mixed-element unstructured meshes, Commun. Comput. Phys., 8 (2010), pp. 585–609. · Zbl 1364.76121
[66] V. A. Titarev and E. F. Toro, ADER schemes for three-dimensional non-linear hyperbolic systems, J. Comput. Phys., 204 (2005), pp. 715–736. · Zbl 1060.65641
[67] E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, New York, 2009. · Zbl 1227.76006
[68] P. Tsoutsanis, V. Titarev, and D. Drikakis, WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J. Comput. Phys., 230 (2011), pp. 1585–1601. · Zbl 1210.65160
[69] R. Wang, H. Feng, and R. J. Spiteri, Observations on the fifth-order WENO method with non-uniform meshes, Appl. Math. Comput., 196 (2008), pp. 433–447. · Zbl 1134.65060
[70] Y. Zhang and C. Shu, Third order WENO scheme on three dimensional tetrahedral meshes, Commun. Comput. Phys., 5 (2009), pp. 836–848. · Zbl 1364.65177
[71] J. Zhu and J. Qiu, A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws, J. Comput. Phys., 318 (2016), pp. 110–121. · Zbl 1349.65365
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.