A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach
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- by Bernardo Cockburn and Pierre-Alain Gremaud PDF
- Math. Comp. 65 (1996), 533-573 Request permission
Abstract:
In this paper, we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original Kuznetsov approximation theory. As a first application of this general technique, we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution. Thus, we obtain optimal error estimates for the Engquist-Osher scheme without using the fact (i) that the solution is uniformly bounded, (ii) that the scheme is total variation diminishing, and (iii) that the discrete semigroup associated with the scheme has the L$^{1}$-contraction property, which guarantees an upper bound for the modulus of continuity in time of the approximate solution.References
- Bernardo Cockburn, Quasimonotone schemes for scalar conservation laws. I, SIAM J. Numer. Anal. 26 (1989), no. 6, 1325–1341. MR 1025091, DOI 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 10.1137/0727017
- 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 10.1137/0727017
- B. Cockburn, F. Coquel, and P. LeFloch, Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), 687–705.
- Bernardo Cockburn, Frédéric Coquel, and Philippe LeFloch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), no. 207, 77–103. MR 1240657, DOI 10.1090/S0025-5718-1994-1240657-4
- B. Cockburn and H. Gau, A posteriori error estimates for general numerical schemes for conservations laws, Mat. Apl. Comput. 14 (1995), 37–47.
- B. Cockburn and P.-A. Gremaud, An error estimate for finite element methods for conservations laws, University of Minnesota Supercomputer Institute Research Report 93-128, SIAM J. Numer. Anal. (to appear).
- 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 10.1090/S0025-5718-1991-1079010-2
- Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, DOI 10.1007/BF00752112
- Björn Engquist and Stanley Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981), no. 154, 321–351. MR 606500, DOI 10.1090/S0025-5718-1981-0606500-X
- T. Geveci, The significance of the stability of difference schemes in different $l^{p}$-spaces, SIAM Rev. 24 (1982), no. 4, 413–426. MR 678560, DOI 10.1137/1024099
- 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 10.1002/cpa.3160290305
- Dietmar Kröner and Mirko Rokyta, Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions, SIAM J. Numer. Anal. 31 (1994), no. 2, 324–343. MR 1276703, DOI 10.1137/0731017
- S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
- N. N. Kuznecov, The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation, Ž. Vyčisl. Mat i Mat. Fiz. 16 (1976), no. 6, 1489–1502, 1627 (Russian). MR 483509
- Bradley J. Lucier, A stable adaptive numerical scheme for hyperbolic conservation laws, SIAM J. Numer. Anal. 22 (1985), no. 1, 180–203. MR 772891, DOI 10.1137/0722012
- 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 10.1137/0722064
- Bradley J. Lucier, On nonlocal monotone difference schemes for scalar conservation laws, Math. Comp. 47 (1986), no. 175, 19–36. MR 842121, DOI 10.1090/S0025-5718-1986-0842121-6
- Bradley J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), no. 173, 59–69. MR 815831, DOI 10.1090/S0025-5718-1986-0815831-4
- Stanley Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21 (1984), no. 2, 217–235. MR 736327, DOI 10.1137/0721016
- Benoît Perthame and Richard Sanders, The Neumann problem for nonlinear second order singular perturbation problems, SIAM J. Math. Anal. 19 (1988), no. 2, 295–311. MR 930028, DOI 10.1137/0519022
- Richard Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91–106. MR 679435, DOI 10.1090/S0025-5718-1983-0679435-6
- 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 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 10.1051/m2an/1991250607491
- Eitan Tadmor and Tamir Tassa, On the piecewise smoothness of entropy solutions to scalar conservation laws, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1631–1652. MR 1239926, DOI 10.1080/03605309308820988
- J.-P. Vila, Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws, Model. Math. Anal. Numer. 28 (1994), 267–295.
- A. I. Vol′pert and S. I. Hudjaev, The Cauchy problem for second order quasilinear degenerate parabolic equations, Mat. Sb. (N.S.) 78 (120) (1969), 374–396 (Russian). MR 0264232
Additional Information
- Bernardo Cockburn
- Affiliation: School of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, Minnesota 55455
- Email: cockburn@math.umn.edu
- Pierre-Alain Gremaud
- Affiliation: Department of Mathematics, North Carolina State University, Box 8205, Raleigh, North Carolina 27695-8205
- Email: gremaud@dali.math.ncsu.edu
- Received by editor(s): August 22, 1994
- Received by editor(s) in revised form: February 22, 1995
- Additional Notes: First author partially supported by the National Science Foundation (Grant DMS-9407952) and by the University of Minnesota Supercomputer Institute.
Second author partially supported by the University of Minnesota Supercomputer Institute. - © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 533-573
- MSC (1991): Primary 65M60, 65N30, 35L65
- DOI: https://doi.org/10.1090/S0025-5718-96-00701-6
- MathSciNet review: 1333308