×

zbMATH — the first resource for mathematics

Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers. (English) Zbl 1349.76309
Summary: In this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by the second author et al. [ibid. 261, Part A, 172–208 (2014; Zbl 1349.76426)] to construct a new class of computationally efficient high order Lagrangian ADER-WENO one-step ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstruction operator allows the algorithm to achieve high order of accuracy in space, while high order of accuracy in time is obtained by the use of an ADER time-stepping technique based on a local space-time Galerkin predictor. The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the grid, considering the entire Voronoi neighborhood of each node and allow for larger time steps than conventional one-dimensional Riemann solvers. The results produced by the multidimensional Riemann solver are then used twice in our one-step ALE algorithm: first, as a node solver that assigns a unique velocity vector to each vertex, in order to preserve the continuity of the computational mesh; second, as a building block for genuinely multidimensional numerical flux evaluation that allows the scheme to run with larger time steps compared to conventional finite volume schemes that use classical one-dimensional Riemann solvers in normal direction. The space-time flux integral computation is carried out at the boundaries of each triangular space-time control volume using the Simpson quadrature rule in space and Gauss-Legendre quadrature in time. A rezoning step may be necessary in order to overcome element overlapping or crossing-over. Since our one-step ALE finite volume scheme is based directly on a space-time conservation formulation of the governing PDE system, the remapping stage is not needed, making our algorithm a so-called direct ALE method. We apply the method presented in this article to two systems of hyperbolic conservation laws, namely the Euler equations of compressible gas dynamics and the equations of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to fourth order of accuracy in space and time have been carried out. Several numerical test problems have been solved to validate the new approach. Furthermore, the new high order Lagrangian schemes based on genuinely multidimensional Riemann solvers have been carefully compared with high order Lagrangian finite volume schemes based on conventional one-dimensional Riemann solvers. It has been clearly shown that due to the less restrictive CFL condition the new schemes based on multidimensional HLL and HLLC Riemann solvers are computationally more efficient than the ones based on a conventional one-dimensional Riemann solver technique.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abgrall, R., Approximation du problème de Riemann vraiment multidimensionnel des équations d’Euler par une méthode de type roe, I: la linéarisation, C.R. Acad. Sci., Ser. I, 319, 499-504, (1994) · Zbl 0813.76074
[2] Abgrall, R., Approximation du problème de Riemann vraiment multidimensionnel des équations d’Euler par une méthode de type roe, II: solution du problème de Riemann approché, C.R. Acad. Sci., Ser. I, 319, 625-629, (1994) · Zbl 0813.76075
[3] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys., 114, 45-58, (1994) · Zbl 0822.65062
[4] Aboiyar, T.; Georgoulis, E. H.; Iske, A., Adaptive ADER methods using kernel-based polyharmonic spline WENO reconstruction, SIAM J. Sci. Comput., 32, 3251-3277, (2010) · Zbl 1221.65236
[5] Balsara, D., Total variation diminishing scheme for relativistic magnetohydrodynamics, Astrophys. J. Suppl. Ser., 132, 83-101, (2001)
[6] Balsara, D., Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151, 149-184, (2004)
[7] Balsara, D.; Spicer, D., Maintaining pressure positivity in magnetohydrodynamic simulations, J. Comput. Phys., 148, 133-148, (1999) · Zbl 0930.76050
[8] Balsara, D.; Spicer, D., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comput. Phys., 149, 270-292, (1999) · Zbl 0936.76051
[9] Balsara, D. S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. Comput. Phys., 174, 614-648, (2001) · Zbl 1157.76369
[10] Balsara, D. S., Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 229, 1970-1993, (2010) · Zbl 1303.76140
[11] Balsara, D. S., A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 231, 7476-7503, (2012) · Zbl 1284.76261
[12] Balsara, D. S., Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics, J. Comput. Phys., 231, 7504-7517, (2012)
[13] Balsara, D. S.; Dumbser, M.; Abgrall, R., Multidimensional HLLC Riemann solver for unstructured meshes, J. Comput. Phys., 261, 172-208, (2014) · Zbl 1349.76426
[14] 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
[15] Balsara, D. S.; Shu, C. W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 405-452, (2000) · Zbl 0961.65078
[16] Barth, T. J.; Frederickson, P. O., Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, (28th Aerospace Sciences Meeting, (January 1990)), AIAA paper no. 90-0013
[17] T.J. Barth, D.C. Jespersen, The design and application of upwind schemes on unstructured meshes, 1989, pp. 1-12, AIAA Paper 89-0366.
[18] Ben-Artzi, M.; Falcovitz, J., A second-order Godunov-type scheme for compressible fluid dynamics, J. Comput. Phys., 55, 1-32, (1984) · Zbl 0535.76070
[19] Berndt, M.; Breil, J.; Galera, S.; Kucharik, M.; Maire, P. H.; Shashkov, M. J., Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods, J. Comput. Phys., 230, 6664-6687, (2011) · Zbl 1408.65077
[20] Billett, S. J.; Toro, E. F., On waf-type schemes for multidimensional hyperbolic conservation laws, J. Comput. Phys., 130, 1-24, (1997) · Zbl 0873.65088
[21] 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
[22] Book, D. L.; Boris, J. P.; Hain, K., Flux-corrected transport II: generalizations of the method, J. Comput. Phys., 18, 248-283, (1975) · Zbl 0306.76004
[23] Boris, J. P.; Book, D. L., Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys., 11, 38-69, (1973) · Zbl 0251.76004
[24] 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
[25] 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, (2014), submitted for publication
[26] Bourgeade, A.; LeFloch, P.; Raviart, P. A., An asymptotic expansion for the solution of the generalized Riemann problem. part II: application to the gas dynamics equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 6, 437-480, (1989) · Zbl 0703.35106
[27] 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
[28] 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
[29] Cesenek, J.; Feistauer, M.; Horacek, J.; Kucera, V.; Prokopova, J., Simulation of compressible viscous flow in time-dependent domains, Appl. Math. Comput., 219, 7139-7150, (2013) · Zbl 1426.76233
[30] 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
[31] 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
[32] Cheng, J.; Shu, C. W., Improvement on spherical symmetry in two-dimensional cylindrical coordinates for a class of control volume Lagrangian schemes, Commun. Comput. Phys., 11, 1144-1168, (2012) · Zbl 1373.76158
[33] 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, 4028-4050, (2011) · Zbl 1218.65091
[34] Cockburn, B.; Karniadakis, G. E.; Shu, C. W., Discontinuous Galerkin methods, Lect. Notes Comput. Sci. Eng., (2000), Springer
[35] Colella, P., A direct Eulerian muscl scheme for gas dynamics, SIAM J. Sci. Stat. Comput., 6, 104-117, (1985) · Zbl 0562.76072
[36] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 171-200, (1990) · Zbl 0694.65041
[37] Dedner, A.; Kemm, F.; Kröner, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys., 175, 645-673, (2002) · Zbl 1059.76040
[38] 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
[39] Després, B.; Mazeran, C., Lagrangian gas dynamics in two-dimensions and Lagrangian systems, Arch. Ration. Mech. Anal., 178, 327-372, (2005) · Zbl 1096.76046
[40] Dubcova, L.; Feistauer, M.; Horacek, J.; Svacek, P., Numerical simulation of interaction between turbulent flow and a vibrating airfoil, Comput. Vis. Sci., 12, 207-225, (2009) · Zbl 1426.74127
[41] Dubiner, M., Spectral methods on triangles and other domains, J. Sci. Comput., 6, 345-390, (1991) · Zbl 0742.76059
[42] Dukowicz, J. K.; Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J. Comput. Phys., 99, 115-134, (1992) · Zbl 0743.76058
[43] Dukowicz, J. K., A general non-iterative Riemann solver for Godunov’s method, J. Comput. Phys., 61, 119-137, (1984) · Zbl 0629.76074
[44] Dumbser, M.; Balsara, D. S.; Toro, E. F.; Munz, C.-D., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227, 8209-8253, (2008) · Zbl 1147.65075
[45] 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
[46] 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
[47] 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
[48] Dumbser, M.; Käser, 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
[49] Dumbser, M.; Toro, E. F., On universal osher-type schemes for general nonlinear hyperbolic conservation laws, Commun. Comput. Phys., 10, 635-671, (2011) · Zbl 1373.76125
[50] Dumbser, M.; Toro, E. F., A simple extension of the osher Riemann solver to non-conservative hyperbolic systems, J. Sci. Comput., 48, 70-88, (2011) · Zbl 1220.65110
[51] 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
[52] Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25, 294-318, (1988) · Zbl 0642.76088
[53] Einfeldt, B.; Munz, C. D.; Roe, P. L.; Sjögreen, B., On Godunov-type methods near low densities, J. Comput. Phys., 92, 273-295, (1991) · Zbl 0709.76102
[54] Feistauer, M.; Horacek, J.; Ruzicka, M.; Svacek, P., Numerical analysis of flow-induced nonlinear vibrations of an airfoil with three degrees of freedom, Comput. Fluids, 49, 110-127, (2011) · Zbl 1271.76165
[55] Feistauer, M.; Kucera, V.; Prokopova, J.; Horacek, J., The ALE discontinuous Galerkin method for the simulatio of air flow through pulsating human vocal folds, AIP Conf. Proc., 1281, 83-86, (2010)
[56] Fey, M., Multidimensional upwinding 1. the method of transport for solving the Euler equations, J. Comput. Phys., 143, 159-180, (1998) · Zbl 0932.76050
[57] Fey, M., Multidimensional upwinding 2. decomposition of the Euler equation into advection equation, J. Comput. Phys., 143, 159-199, (1998)
[58] Le Floch, P.; Raviart, P. A., An asymptotic expansion for the solution of the generalized Riemann problem. part I: general theory, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 5, 179-207, (1988) · Zbl 0679.35064
[59] 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
[60] Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys., 144, 194-212, (1998) · Zbl 1392.76048
[61] 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
[62] 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
[63] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 289-315, (1983) · Zbl 0565.65051
[64] 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
[65] Hirt, C.; Amsden, A.; Cook, J., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys., 14, 227-253, (1974) · Zbl 0292.76018
[66] 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
[67] Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 202-228, (1996) · Zbl 0877.65065
[68] Kamm, J. R.; Timmes, F. X., On efficient generation of numerically robust sedov solutions, (2007), Technical Report LA-UR-07-2849, LANL
[69] Karniadakis, G. E.; Sherwin, S. J., Spectral/hp element methods in CFD, (1999), Oxford University Press · Zbl 0954.76001
[70] Käser, M.; Iske, A., Ader schemes on adaptive triangular meshes for scalar conservation laws, J. Comput. Phys., 205, 486-508, (2005) · Zbl 1072.65116
[71] Kidder, R. E., Laser-driven compression of hollow shells: power requirements and stability limitations, Nucl. Fusion, 1, 3-14, (1976)
[72] 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
[73] 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
[74] 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
[75] Liska, R.; Shashkov, M. J.; Váchal, P.; Wendroff, B., Synchronized flux corrected remapping for ALE methods, Comput. Fluids, 46, 312-317, (2011) · Zbl 1433.76135
[76] 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
[77] 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
[78] Loubère, R.; Maire, P. H.; Shashkov, M. J., Reale: A reconnection arbitrary-Lagrangian-Eulerian method in cylindrical geometry, Comput. Fluids, 46, 59-69, (2011) · Zbl 1305.76066
[79] Lukacova-Medvidova, M.; Morton, K. W.; Warnecke, G., Finite volume evolution Galerkin methods for Euler equations of gas dynamics, Int. J. Numer. Methods Fluids, 40, 425-434, (2002) · Zbl 1023.76026
[80] Lukacova-Medvidova, M.; Morton, K. W.; Warnecke, G., Finite volume evolution Galerkin methods for hyperbolic systems, SIAM J. Sci. Comput., 26, 1-30, (2005) · Zbl 1078.65562
[81] Maire, P. H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J. Comput. Phys., 228, 2391-2425, (2009) · Zbl 1156.76434
[82] 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
[83] 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
[84] 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
[85] Munz, C. D., On Godunov-type schemes for Lagrangian gas dynamics, SIAM J. Numer. Anal., 31, 17-42, (1994) · Zbl 0796.76057
[86] Noh, W. F., Errors for calculations of strong shocks using artificial viscosity and an artificial heat flux, J. Comput. Phys., 72, 78-120, (1987) · Zbl 0619.76091
[87] Olliver-Gooch, C.; Van Altena, M., A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation, J. Comput. Phys., 181, 729-752, (2002) · Zbl 1178.76251
[88] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comput., 38, 339-374, (1982) · Zbl 0483.65055
[89] 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
[90] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[91] Rumsey, C. B.; van Leer, B.; Roe, P. L., A multidimensional flux function with application to the Euler and Navier-Stokes equations, J. Comput. Phys., 105, 306-323, (1993) · Zbl 0767.76039
[92] Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys. USSR, 1, 267-305, (1961)
[93] 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)
[94] 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
[95] Scovazzi, G., Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate variational multiscale approach, J. Comput. Phys., 231, 8029-8069, (2012)
[96] Smith, R. W., AUSM(ALE): a geometrically conservative arbitrary Lagrangian-Eulerian flux splitting scheme, J. Comput. Phys., 150, 268-286, (1999) · Zbl 0936.76046
[97] Stroud, A. H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Inc. Englewood Cliffs, New Jersey · Zbl 0379.65013
[98] Titarev, V. A.; Toro ADER, E. F., Arbitrary high order Godunov approach, J. Sci. Comput., 17, 1-4, 609-618, (December 2002)
[99] Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J. Comput. Phys., 204, 715-736, (2005) · Zbl 1060.65641
[100] Titarev, V. A.; Tsoutsanis, P.; Drikakis, D., WENO schemes for mixed-element unstructured meshes, Commun. Comput. Phys., 8, 585-609, (2010) · Zbl 1364.76121
[101] Toro, E. F.; Titarev, V. A., Derivative Riemann solvers for systems of conservation laws and ADER methods, J. Comput. Phys., 212, 1, 150-165, (2006) · Zbl 1087.65590
[102] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: A practical introduction, (2009), Springer · Zbl 1227.76006
[103] Toro, E. F.; Spruce, M.; Speares, W., Restoration of contact surface in the HLL Riemann solver, Shock Waves, 4, 25-34, (1994) · Zbl 0811.76053
[104] Tsoutsanis, P.; Titarev, V. A.; Drikakis, D., WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J. Comput. Phys., 230, 1585-1601, (2011) · Zbl 1210.65160
[105] 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
[106] 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
[107] 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.