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A priori error estimates for numerical methods for scalar conservation laws. part ii: flux-splitting monotone schemes on irregular Cartesian grids

Authors: Bernardo Cockburn and Pierre-Alain Gremaud
Journal: Math. Comp. 66 (1997), 547-572
MSC (1991): Primary 65M60, 65N30, 35L65
MathSciNet review: 1408372
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Abstract: This paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of $(\Delta x)^{1/2}$ in $L^{\infty }(L^{1})$ for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution. We then extend this result to less consistent schemes, called $p-$consistent schemes, and prove that they converge to the entropy solution with the rate of $(\Delta x)^{\min \{1/2,p\}}$ in $L^{\infty }(L^{1})$; again, no regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of $(\Delta x)^{1/2}$ in $L^{\infty }(L^{1})$. We show that this well-known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.

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  • 1. M.J. Berger, R.J. Leveque, and L.G. Stern, Finite volume methods for irregular one-dimensional grids, Mathematics of Computation 1943-1993: a half century of computational mathematics (Vancouver, BC, 1993), Proc. Sympo. Appl. Math. 48, Amer. Math. Soc., 1994, pp. 255-259. CMP 95:07
  • 2. B. Cockburn, F. Coquel, and P. LeFloch, An error estimate for finite volume methods for conservations laws, Math. Comp. 63 (1994), 77-103. MR 95d:65078
  • 3. B. Cockburn and P.-A. Gremaud, An error estimate for finite element methods for conservations laws, SIAM J. Numer. Anal. 33 (1996), 522-554. CMP 96:12
  • 4. -, A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach, Math. Comp. 65 (1996), 533-573. MR 96g:65089
  • 5. B. Engquist and S. Osher, One sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981), 321-351. MR 82c:65056
  • 6. B. García-Archilla, A supraconvergent scheme for the Korteweg-de-Vries equation, Numer. Math. 61 (1992), 291-310. MR 92k:65146
  • 7. B. García-Archilla and J.M. Sanz-Serna, A finite difference formula for the discretization of $d^{3}/dx^{3}$ on nonuniform grids, Math. Comp. 57 (1991), 239-257. MR 91j:65143
  • 8. A. Harten, B. Engquist, S. Osher and S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys. 71 (1987), 231-303. MR 90a:65199
  • 9. J.D. Hoffman, Relationship between the truncation error of centered finite difference approximations on uniform and nonuniform meshes, J. Comput. Phys. 46 (1982), 469-474. MR 83j:65031
  • 10. H.-O. Kreiss, T.A. Manteuffel, B. Swartz, B. Wendroff, and A.B. White, Jr., Supra-convergent schemes on Irregular grids, Math. Comp. 47 (1986), 537-554. MR 88b:65082
  • 11. S.N. Kru\v{z}kov, First order quasilinear equations in several independent variables, Math. USSR Sbornik 10 (1970), 217-243. MR 42:2159
  • 12. N.N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comp. Math. and Math. Phys. 16 (1976), 105-119. MR 58:3510
  • 13. B. J. Lucier, A stable adaptive scheme for hyperbolic conservation laws, SIAM J. Numer. Anal. 22 (1985), 180-203. MR 86d:65123
  • 14. -, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), 1074-1081. MR 88a:65104
  • 15. -, On nonlocal monotone difference schemes for scalar conservation laws, Math. Comp. 47 (1986), 19-36. MR 87j:65110
  • 16. -, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), 59-69. MR 87m:65141
  • 17. T.A. Manteuffel and A.B. White, Jr., The numerical solution of second-order boundary value problems on nonuniform meshes, Math. Comp. 47 (1986), 511-535. MR 87m:65116
  • 18. H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal. 29 (1992), 1505-1519. MR 93j:65139
  • 19. H. Nessyahu, E. Tadmor and T. Tassa, The convergence rate of Godunov type schemes, SIAM J. Numer. Anal. 31 (1994), 1-16. MR 94m:65140
  • 20. S. Noelle, A note on entropy inequalities and error estimates for higher-order accurate finite volume schemes on irregular grids, Math. Comp. 65 (1996), 1155-1163. MR 96j:65089
  • 21. J. Pike, Grid adaptive algorithms for the solution of the Euler equations on irregular grids, J. Comput. Phys. 71 (1987), 194-223. MR 88g:65083
  • 22. R. Sanders, On Convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), 91-106. MR 84a:65075
  • 23. G. Strang, Accurate partial difference methods II: non-linear problems, Numer. Math. 6 (1964), 37-46. MR 29:4215
  • 24. E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), 891-906. MR 92d:35190
  • 25. E. Turkel, Accuracy of schemes with nonuniform meshes for compressible fluid flow, App. Numer. Math. 2 (1986), 529-550. MR 88m:65137
  • 26. 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. MR 96d:65150
  • 27. B. Wendroff and A.B. White, Jr., Some supraconvergent schemes for hyperbolic equations on irregular grids, Second International Conference on Hyperbolic Problems, Aachen (1988), 671-677. CMP 21:10
  • 28. B. Wendroff and A.B. White, Jr., A supraconvergent scheme for nonlinear hyperbolic systems, Comput. Math. Appl. 18 (1989), 761-767. MR 90g:65121

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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, Minnesota 55455

Pierre-Alain Gremaud
Affiliation: Center for Research in Scientific Computation and Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

Keywords: A priori error estimates, irregular grids, monotone schemes, conservation laws, supraconvergence
Received by editor(s): November 27, 1995
Received by editor(s) in revised form: May 6, 1996
Additional Notes: The first author was partially supported by the National Science Foundation (Grant DMS-9407952) and by the University of Minnesota Supercomputer Institute.
The second author was partially supported by the Army Research Office through grant DAAH04-95-1-0419.
Article copyright: © Copyright 1997 American Mathematical Society

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