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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 in 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 consistent schemes, and prove that they converge to the entropy solution with the rate of in ; 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 in . 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.

**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.**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**, 10.1090/S0025-5718-1994-1240657-4**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.**Bernardo Cockburn and Pierre-Alain Gremaud,*A priori error estimates for numerical methods for scalar conservation laws. I. The general approach*, Math. Comp.**65**(1996), no. 214, 533–573. MR**1333308**, 10.1090/S0025-5718-96-00701-6**5.**Björn Engquist and Stanley Osher,*One-sided difference approximations for nonlinear conservation laws*, Math. Comp.**36**(1981), no. 154, 321–351. MR**606500**, 10.1090/S0025-5718-1981-0606500-X**6.**Bosco García-Archilla,*A supraconvergent scheme for the Korteweg-de Vries equation*, Numer. Math.**61**(1992), no. 3, 291–310. MR**1151772**, 10.1007/BF01385511**7.**B. García-Archilla and J. M. Sanz-Serna,*A finite difference formula for the discretization of 𝑑³/𝑑𝑥³ on nonuniform grids*, Math. Comp.**57**(1991), no. 195, 239–257. MR**1079016**, 10.1090/S0025-5718-1991-1079016-3**8.**Ami Harten, Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy,*Uniformly high-order accurate essentially nonoscillatory schemes. III*, J. Comput. Phys.**71**(1987), no. 2, 231–303. MR**897244**, 10.1016/0021-9991(87)90031-3**9.**Joe D. Hoffman,*Relationship between the truncation errors of centered finite-difference approximations on uniform and nonuniform meshes*, J. Comput. Phys.**46**(1982), no. 3, 469–474. MR**673711**, 10.1016/0021-9991(82)90028-6**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), no. 176, 537–554. MR**856701**, 10.1090/S0025-5718-1986-0856701-5**11.**S. N. Kružkov,*First order quasilinear equations with several independent variables.*, Mat. Sb. (N.S.)**81 (123)**(1970), 228–255 (Russian). MR**0267257****12.**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**0483509****13.**Bradley J. Lucier,*A stable adaptive numerical scheme for hyperbolic conservation laws*, SIAM J. Numer. Anal.**22**(1985), no. 1, 180–203. MR**772891**, 10.1137/0722012**14.**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**, 10.1137/0722064**15.**Bradley J. Lucier,*On nonlocal monotone difference schemes for scalar conservation laws*, Math. Comp.**47**(1986), no. 175, 19–36. MR**842121**, 10.1090/S0025-5718-1986-0842121-6**16.**Bradley J. Lucier,*A moving mesh numerical method for hyperbolic conservation laws*, Math. Comp.**46**(1986), no. 173, 59–69. MR**815831**, 10.1090/S0025-5718-1986-0815831-4**17.**Thomas A. Manteuffel and Andrew B. White Jr.,*The numerical solution of second-order boundary value problems on nonuniform meshes*, Math. Comp.**47**(1986), no. 176, 511–535, S53–S55. MR**856700**, 10.1090/S0025-5718-1986-0856700-3**18.**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**, 10.1137/0729087**19.**Haim Nessyahu, Eitan Tadmor, and Tamir Tassa,*The convergence rate of Godunov type schemes*, SIAM J. Numer. Anal.**31**(1994), no. 1, 1–16. MR**1259963**, 10.1137/0731001**20.**Sebastian Noelle,*A note on entropy inequalities and error estimates for higher-order accurate finite volume schemes on irregular families of grids*, Math. Comp.**65**(1996), no. 215, 1155–1163. MR**1344618**, 10.1090/S0025-5718-96-00737-5**21.**J. Pike,*Grid adaptive algorithms for the solution of the Euler equations on irregular grids*, J. Comput. Phys.**71**(1987), no. 1, 194–223. MR**895526**, 10.1016/0021-9991(87)90027-1**22.**Richard Sanders,*On convergence of monotone finite difference schemes with variable spatial differencing*, Math. Comp.**40**(1983), no. 161, 91–106. MR**679435**, 10.1090/S0025-5718-1983-0679435-6**23.**Gilbert Strang,*Accurate partial difference methods. II. Non-linear problems*, Numer. Math.**6**(1964), 37–46. MR**0166942****24.**Eitan Tadmor,*Local error estimates for discontinuous solutions of nonlinear hyperbolic equations*, SIAM J. Numer. Anal.**28**(1991), no. 4, 891–906. MR**1111445**, 10.1137/0728048**25.**Eli Turkel,*Accuracy of schemes with nonuniform meshes for compressible fluid flows*, Appl. Numer. Math.**2**(1986), no. 6, 529–550. MR**871091**, 10.1016/0168-9274(86)90006-1**26.**J.-P. Vila,*Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes*, RAIRO Modél. Math. Anal. Numér.**28**(1994), no. 3, 267–295 (English, with English and French summaries). MR**1275345****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.**Burton Wendroff and Andrew B. White Jr.,*A supraconvergent scheme for nonlinear hyperbolic systems*, Comput. Math. Appl.**18**(1989), no. 8, 761–767. MR**1009864**, 10.1016/0898-1221(89)90232-0

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65M60,
65N30,
35L65

Retrieve articles in all journals with MSC (1991): 65M60, 65N30, 35L65

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

DOI:
https://doi.org/10.1090/S0025-5718-97-00838-7

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