We present a (partial) historical summary of the mathematical analysis of finite difference and finite volume methods, paying special attention to the Lax–Richtmyer and Lax–Wendroff theorems. We then state a Lax–Wendroff consistency result for convection operators on staggered grids (often used in fluid flow simulations), which illustrates a recent generalization of the flux consistency notion designed to cope with general discrete functions.
Revised:
Accepted:
Online First:
Robert Eymard 1; Thierry Gallouët 2; Raphaele Herbin 2; Jean-Claude Latché 3
CC-BY 4.0
@article{CRMECA_2022__350_S1_A9_0,
author = {Robert Eymard and Thierry Gallou\"et and Raphaele Herbin and Jean-Claude Latch\'e},
title = {Finite volume schemes and {Lax{\textendash}Wendroff} consistency},
journal = {Comptes Rendus. M\'ecanique},
publisher = {Acad\'emie des sciences, Paris},
year = {2022},
doi = {10.5802/crmeca.132},
language = {en},
note = {Online first},
}
TY - JOUR AU - Robert Eymard AU - Thierry Gallouët AU - Raphaele Herbin AU - Jean-Claude Latché TI - Finite volume schemes and Lax–Wendroff consistency JO - Comptes Rendus. Mécanique PY - 2022 PB - Académie des sciences, Paris N1 - Online first DO - 10.5802/crmeca.132 LA - en ID - CRMECA_2022__350_S1_A9_0 ER -
Robert Eymard; Thierry Gallouët; Raphaele Herbin; Jean-Claude Latché. Finite volume schemes and Lax–Wendroff consistency. Comptes Rendus. Mécanique, Online first (2022), pp. 1-13. doi : 10.5802/crmeca.132.
[1] Survey of the stability of linear finite difference equations, Commun. Pure Appl. Math., Volume 9 (1956) no. 2, pp. 267-293 | DOI | MR
[2] Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia, 2004
[3] Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002 | DOI
[4] Des mathématiciens découvrent les volumes finis, Matapli, Volume 28 (1991), pp. 37-48
[5] Finite volume methods, Techniques of Scientific Computing (Part 3) (J. L. Lions; P. Ciarlet, eds.) (Handbook of Numerical Analysis), Volume 7, Elsevier, Amsterdam, 2000, pp. 713-1020
[6] Systems of conservation laws, Commun. Pure Appl. Math., Volume 13 (1960), pp. 217-237 | DOI | Zbl
[7] On the weak consistency of finite volume schemes for conservation laws on general meshes, SeMA J., Volume 76 (2019), pp. 581-594 | DOI | MR | Zbl
[8] Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., Volume 100 (1928) no. 1, pp. 32-74 voir IBM J. Res. Dev. 11 (1967), p. 215–234, pour une version en anglais | DOI | Zbl
[9] Finite volume schemes for diffusion equations: introduction to and review of modern methods, Math. Models Methods Appl. Sci., Volume 24 (2014) no. 8, pp. 1575-1619 | DOI | MR | Zbl
[10] Monotone difference approximations for a first order quasilinear equation, Dokl. Akad. Nauk SSSR, Volume 229 (1976) no. 6, pp. 1317-1320 | MR
[11] Monotone difference approximations for scalar conservation laws, Math. Comput., Volume 34 (1980) no. 149, pp. 1-21 | DOI | MR | Zbl
[12] On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. Numer. Anal., Volume 21 (1984) no. 1, pp. 1-23 (With an appendix by Peter D. Lax) | DOI | MR
[13] Convergence d’un schéma décentré amont sur un maillage triangulaire pour un problème hyperbolique linéaire, Modélisation mathématique et analyse numérique, Volume 26 (1992) no. 7, pp. 835-853 | Zbl
[14] Mesure, intégration, probabilités, Ellipses, Paris, 2013 https://hal.archives-ouvertes.fr/cel-00637007v2
[15] First order quasilinear equations with several independent variables, Math. USSR Sb. (N.S.), Volume 81 (1970) no. 123, pp. 228-255 | MR
[16] Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal., Volume 88 (1985) no. 3, pp. 223-270 | DOI | MR | Zbl
[17] Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chin. Ann. Math. Ser. B, Volume 16 (1995) no. 1, pp. 1-14 A Chinese summary appears in Chin. Ann. Math. Ser. A 16 (1995), no. 1, p. 119 | MR | Zbl
[18] Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal., Volume 32 (1995) no. 3, pp. 687-705 | DOI | MR | Zbl
[19] Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, Volume 8 (1965), pp. 2182-2189 | DOI | MR
[20] A potential enstrophy and energy conserving scheme for the shallow water equations, Mon. Weather Rev., Volume 109 (1981), pp. 18-36 | DOI
[21] Numerical Heat Transfer and Fluid Flow, Series in Computational Methods in Mechanics and Thermal Sciences, XIII, Hemisphere Publishing Corporation, McGraw-Hill Book Company, Washington, New York, London, 1980, 197 pages
[22] Convergence of the Marker-And-Cell scheme for the incompressible Navier–Stokes equations on non-uniform grids, Found. Comput. Math., Volume 18 (2018) no. 1, pp. 249-289 | DOI | MR | Zbl
[23] Error estimates for the implicit MAC scheme for the compressible Navier–Stokes equations, Numer. Math., Volume 141 (2019) no. 2, pp. 495-567 | DOI | MR | Zbl
[24] Low Mach number limit of some staggered schemes for compressible barotropic flows, Math. Comput., Volume 90 (2021) no. 329, pp. 1039-1087 | DOI | MR | Zbl
[25] Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations, ESAIM: Math. Model. Numer. Anal., Volume 52 (2018) no. 3, pp. 893-944 | DOI | MR | Zbl
[26] Conservativity and weak consistency of a class of staggered finite volume methods for the Euler equations, Math. Comput., Volume 90 (2021) no. 329, pp. 1155-1177 | DOI | MR | Zbl
[27] Lax–Wendroff consistency of finite volume schemes for systems of non linear conservation laws: extension to staggered schemes, SeMA J., Volume 79 (2022) no. 2, pp. 333-354 | DOI | MR | Zbl
[28] A consistent quasi–second-order staggered scheme for the two-dimensional shallow water equations, IMA J. Numer. Anal. (2021), drab086 | DOI
[29] A software components library for the computation of reactive turbulent flows https://gforge.irsn.fr/gf/project/isis
Cited by Sources:
Comments - Policy