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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
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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.


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  • [1] G.-Q. Chen, Q. Du, and E. Tadmor, Spectral viscosity approximation to multidimensional scalar conservation laws, Math. Comp. 61 (1993), 629-643. MR 1185240 (94b:35168)
  • [2] 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.
  • [3] -, Entropy flux-splittings for hyperbolic conservation laws. Part 2: Gas dynamics equations, in preparation.
  • [4] I.L. 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), 815-844. MR 1003436 (90k:65157)
  • [5] B. Cockburn, The quasi-monotone schemes for scalar conservation laws. I, SIAM J. Numer. Anal. 26 (1989), 1325-1341. MR 1025091 (91b:65106)
  • [6] -, The quasi-monotone schemes for scalar conservation laws. II, SIAM J. Numer. Anal. 27 (1990), 247-258. MR 1034933 (91b:65107)
  • [7] -, The quasi-monotone schemes for scalar conservation laws. III, SIAM J. Numer. Anal. 27 (1990), 259-276.
  • [8] -, On the continuity in $ BV(\Omega )$ of the $ {L^2}$-projection into finite element spaces, Math. Comp. 57 (1991), 551-561. MR 1094943 (92a:65288)
  • [9] B. Cockburn, F. Coquel and Ph. LeFloch, Convergence of finite volume methods for multidimensional conservation laws, SIAM J. Numer. Anal. (to appear). MR 1335651 (97f:65051)
  • [10] B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comp. 52 (1989), 411-435. MR 983311 (90k:65160)
  • [11] B. Cockburn, S.-C. Hou, and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comp. 54 (1990), 545-581. MR 1010597 (90k:65162)
  • [12] 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.
  • [13] F. Coquel and Ph. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions, C. R. Acad. Sci. Paris, Série I, 310 17 (1990), 455-460. MR 1046532 (91d:65131)
  • [14] -, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp. 57 (1991), 169-210. MR 1079010 (91m:65229)
  • [15] -, Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory, SIAM J. Numer. Anal. 30 (1993), 675-700. MR 1220646 (94e:65092)
  • [16] M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), 1-21. MR 551288 (81b:65079)
  • [17] R.J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27-70. MR 684413 (84k:35091)
  • [18] -, Measure-valued solutions to conservations laws, Arch. Rational Mech. Anal. 88 (1985), 223-270. MR 775191 (86g:35121)
  • [19] J.B. Goodman and R.J. LeVeque, On the accuracy of stable schemes for 2D scalar conservation laws, Math. Comp. 45 (1985), 15-21. MR 790641 (86f:65149)
  • [20] A. Harten, J.M. Hyman, and P.D. Lax, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), 297-322. MR 0413526 (54:1640)
  • [21] A. Harten, P.D. Lax, and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), 35-61. MR 693713 (85h:65188)
  • [22] 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.
  • [23] T.Y. Hou and Ph.G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comp. 62 (1994), 497-530. MR 1201068 (94g:65093)
  • [24] C. Johnson and A. Szepessy, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1988), 427-444. MR 906180 (88h:65164)
  • [25] -, A posteriori error estimate for a finite element method, Preprint (1993).
  • [26] S.N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243.
  • [27] -, 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)
  • [28] 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.
  • [29] -, 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.
  • [30] P.D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR 0093653 (20:176)
  • [31] -, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, Philadelphia, PA, 1973.
  • [32] 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).
  • [33] Ph. LeFloch and J.G. Liu, Entropy and monotonicity consistent EMO schemes for conservation laws, in preparation.
  • [34] B.J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), 1074-1081. MR 811184 (88a:65104)
  • [35] -, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), 59-69. MR 815831 (87m:65141)
  • [36] -, On nonlocal monotone difference schemes for scalar conservation laws, Math. Comp. 47 (1986), 19-36. MR 842121 (87j:65110)
  • [37] S. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal. 29 (1992), 1-15. MR 1191133 (93j:65139)
  • [38] S. Osher, Riemann solvers, the entropy condition and difference approximations, SIAM J. Numer. Anal. 21 (1984), 217-235. MR 736327 (86d:65119)
  • [39] S. Osher and R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comp. 41 (1983), 321-336. MR 717689 (85i:65121)
  • [40] S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws, Math. Comp. 50 (1988), 19-51. MR 917817 (89m:65086)
  • [41] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), 91-106. MR 679435 (84a:65075)
  • [42] -, Finite difference techniques for nonlinear hyperbolic conservation laws, Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985, pp. 209-220. MR 818789 (87d:65103)
  • [43] J.S. Smoller, Shock waves and reaction diffusion equations, Springer-Verlag, New York, 1983. MR 688146 (84d:35002)
  • [44] A. Szepessy, Convergence of a shock-capturing streamline diffusion finite element method for scalar conservation laws in two space dimensions, Math. Comp. 53 (1989), 527-545. MR 979941 (90h:65156)
  • [45] -, Convergence of a streamline diffusion finite element method for a conservation law with boundary conditions, RAIRO Modél. Math. Anal. Numér. 25 (1991), 749-783. MR 1135992 (92g:65115)
  • [46] E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp. 43 (1984), 369-382. MR 758189 (86g:65163)
  • [47] -, Semi-discrete approximations to nonlinear systems of conservation laws; consistency and $ {L^\infty }$ imply convergence, ICASE Report 88-41 (1988).
  • [48] -, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), 891-906. MR 1111445 (92d:35190)
  • [50] J.P. Vila, Problèmes nonlinéaires appliqués, Ecoles CEA-EDF-INRIA, Clamart, France, 1993.
  • [51] A.I. Volpert, The space BV and quasilinear equations, Math. USSR Sb. 2 (1967), 257-267. MR 0216338 (35:7172)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1994-1240657-4
Keywords: Multidimensional conservation law, discontinuous solution, finite volume method, error estimate
Article copyright: © Copyright 1994 American Mathematical Society

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