An error estimate for finite volume methods for multidimensional conservation laws
Authors:
Bernardo Cockburn, Frédéric Coquel and Philippe LeFloch
Journal:
Math. Comp. 63 (1994), 77-103
MSC:
Primary 65M15; Secondary 35L65, 65M60
DOI:
https://doi.org/10.1090/S0025-5718-1994-1240657-4
MathSciNet review:
1240657
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, an ${L^\infty }({L^1})$-error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained. These methods can be formally high-order accurate and are defined on general triangulations. The error is proven to be of order ${h^{1/4}}$, where h represents the "size" of the mesh, via an extension of Kuznetsov approximation theory for which no estimate of the total variation and of the modulus of continuity in time are needed. The result is new even for the finite volume method constructed from monotone numerical flux functions.
- Gui Qiang Chen, Qiang Du, and Eitan Tadmor, Spectral viscosity approximations to multidimensional scalar conservation laws, Math. Comp. 61 (1993), no. 204, 629–643. MR 1185240, DOI https://doi.org/10.1090/S0025-5718-1993-1185240-3 G.-Q. Chen and Ph. LeFloch, Entropy flux-splittings for hyperbolic conservation laws. Part 1: general framework, submitted to Comm. Pure Appl. Math., in preparation. ---, Entropy flux-splittings for hyperbolic conservation laws. Part 2: Gas dynamics equations, in preparation.
- I-Liang Chern, Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities, Comm. Pure Appl. Math. 42 (1989), no. 6, 815–844. MR 1003436, DOI https://doi.org/10.1002/cpa.3160420606
- Bernardo Cockburn, Quasimonotone schemes for scalar conservation laws. I, SIAM J. Numer. Anal. 26 (1989), no. 6, 1325–1341. MR 1025091, DOI https://doi.org/10.1137/0726077
- Bernardo Cockburn, Quasimonotone schemes for scalar conservation laws. II, III, SIAM J. Numer. Anal. 27 (1990), no. 1, 247–258, 259–276. MR 1034933, DOI https://doi.org/10.1137/0727017 ---, The quasi-monotone schemes for scalar conservation laws. III, SIAM J. Numer. Anal. 27 (1990), 259-276.
- Bernardo Cockburn, On the continuity in ${\rm BV}(\Omega )$ of the $L^2$-projection into finite element spaces, Math. Comp. 57 (1991), no. 196, 551–561. MR 1094943, DOI https://doi.org/10.1090/S0025-5718-1991-1094943-9
- B. Cockburn, F. Coquel, and P. G. LeFloch, Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), no. 3, 687–705. MR 1335651, DOI https://doi.org/10.1137/0732032
- Bernardo Cockburn and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp. 52 (1989), no. 186, 411–435. MR 983311, DOI https://doi.org/10.1090/S0025-5718-1989-0983311-4
- Bernardo Cockburn, Suchung Hou, and Chi-Wang Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp. 54 (1990), no. 190, 545–581. MR 1010597, DOI https://doi.org/10.1090/S0025-5718-1990-1010597-0 B. Cockburn and C.-W. Shu, The Runge-Kutta local projection ${P^1}$ discontinuous Galerkin finite element method for scalar conservation laws, ICASE Report No. 91-32, 1991.
- Frédéric Coquel and Philippe LeFloch, Convergence de schémas aux différences finies pour des lois de conservation à plusieurs dimensions d’espace, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 6, 455–460 (French, with English summary). MR 1046532
- Frédéric Coquel and Philippe LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp. 57 (1991), no. 195, 169–210. MR 1079010, DOI https://doi.org/10.1090/S0025-5718-1991-1079010-2
- Frédéric Coquel and Philippe LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory, SIAM J. Numer. Anal. 30 (1993), no. 3, 675–700. MR 1220646, DOI https://doi.org/10.1137/0730033
- Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, DOI https://doi.org/10.1090/S0025-5718-1980-0551288-3
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, DOI https://doi.org/10.1007/BF00251724
- Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, DOI https://doi.org/10.1007/BF00752112
- Jonathan B. Goodman and Randall J. LeVeque, On the accuracy of stable schemes for $2$D scalar conservation laws, Math. Comp. 45 (1985), no. 171, 15–21. MR 790641, DOI https://doi.org/10.1090/S0025-5718-1985-0790641-4
- A. Harten, J. M. Hyman, and P. D. Lax, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR 413526, DOI https://doi.org/10.1002/cpa.3160290305
- Amiram Harten, Peter D. Lax, and Bram van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), no. 1, 35–61. MR 693713, DOI https://doi.org/10.1137/1025002 D. Hoff and J.S. Smoller, Error bounds for the Glimm scheme for a scalar conservation law, Trans. Amer. Math. Soc. 289 (1988), 611-642.
- Thomas Y. Hou and Philippe G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comp. 62 (1994), no. 206, 497–530. MR 1201068, DOI https://doi.org/10.1090/S0025-5718-1994-1201068-0
- Claes Johnson and Anders Szepessy, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1987), no. 180, 427–444. MR 906180, DOI https://doi.org/10.1090/S0025-5718-1987-0906180-5 ---, A posteriori error estimate for a finite element method, Preprint (1993). S.N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243. ---, On the methods of construction of the general solution of the Cauchy problem for first order quasilinear equations, Uspehi Mat. Nauk 20 (1965), 112-118. (Russian) N.N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comput. Math. and Math. Phys. 16 (1976), 105-119. ---, On stable methods for solving nonlinear first-order partial differential equations in the class of discontinuous solutions, Topics in Numerical Analysis III (Proc. Roy. Irish Acad. Conf.), Trinity College, Dublin, 1976, pp. 183-192.
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI https://doi.org/10.1002/cpa.3160100406 ---, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, Philadelphia, PA, 1973. Ph. LeFloch, Convergence des méthodes de volumes finis monotones pour les lois de conservation scalaires, Communication to Ecole CEA-EDF-INRIA, Meeting on finite volume methods, October 1992 (unpublished notes). Ph. LeFloch and J.G. Liu, Entropy and monotonicity consistent EMO schemes for conservation laws, in preparation.
- Bradley J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), no. 6, 1074–1081. MR 811184, DOI https://doi.org/10.1137/0722064
- Bradley J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), no. 173, 59–69. MR 815831, DOI https://doi.org/10.1090/S0025-5718-1986-0815831-4
- Bradley J. Lucier, On nonlocal monotone difference schemes for scalar conservation laws, Math. Comp. 47 (1986), no. 175, 19–36. MR 842121, DOI https://doi.org/10.1090/S0025-5718-1986-0842121-6
- Haim Nessyahu and Eitan Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal. 29 (1992), no. 6, 1505–1519. MR 1191133, DOI https://doi.org/10.1137/0729087
- Stanley Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21 (1984), no. 2, 217–235. MR 736327, DOI https://doi.org/10.1137/0721016
- Stanley Osher and Richard Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comp. 41 (1983), no. 164, 321–336. MR 717689, DOI https://doi.org/10.1090/S0025-5718-1983-0717689-8
- Stanley Osher and Eitan Tadmor, On the convergence of difference approximations to scalar conservation laws, Math. Comp. 50 (1988), no. 181, 19–51. MR 917817, DOI https://doi.org/10.1090/S0025-5718-1988-0917817-X
- Richard Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91–106. MR 679435, DOI https://doi.org/10.1090/S0025-5718-1983-0679435-6
- Richard Sanders, Finite difference techniques for nonlinear hyperbolic conservation laws, Large-scale computations in fluid mechanics, Part 2 (La Jolla, Calif., 1983) Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985, pp. 209–220. MR 818789
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
- Anders Szepessy, Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions, Math. Comp. 53 (1989), no. 188, 527–545. MR 979941, DOI https://doi.org/10.1090/S0025-5718-1989-0979941-6
- A. Szepessy, Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 6, 749–782 (English, with French summary). MR 1135992, DOI https://doi.org/10.1051/m2an/1991250607491
- Eitan Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp. 43 (1984), no. 168, 369–381. MR 758189, DOI https://doi.org/10.1090/S0025-5718-1984-0758189-X ---, Semi-discrete approximations to nonlinear systems of conservation laws; consistency and ${L^\infty }$ imply convergence, ICASE Report 88-41 (1988).
- Eitan Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), no. 4, 891–906. MR 1111445, DOI https://doi.org/10.1137/0728048 J.P. Vila, Problèmes nonlinéaires appliqués, Ecoles CEA-EDF-INRIA, Clamart, France, 1993.
- A. I. Vol′pert, Spaces ${\rm BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338
Retrieve articles in Mathematics of Computation with MSC: 65M15, 35L65, 65M60
Retrieve articles in all journals with MSC: 65M15, 35L65, 65M60
Additional Information
Keywords:
Multidimensional conservation law,
discontinuous solution,
finite volume method,
error estimate
Article copyright:
© Copyright 1994
American Mathematical Society
