Mathematics of Computation

On Wavewise Entropy Inequalities for High-Resolution Schemes. I: The Semidiscrete Case

Author(s): Huanan Yang.
Journal: Math. Comp. 65 (1996), 45-67.
MSC (1991): Primary 65M60, 65M12, 35L65
Supplement: Additional information related to this article.
Retrieve article in: PDF
This article is available free of charge

Abstract: We develop a new approach, the method of wavewise entropy inequalities for the numerical analysis of hyperbolic conservation laws. The method is based on a new extremum tracking theory and Volpert's theory of BV solutions. The method yields a sharp convergence criterion which is used to prove the convergence of generalized MUSCL schemes and a class of schemes using flux limiters previously discussed in 1984 by Sweby.

1: F. Bouchut, C. Bourdarias, and B. Perthame, Un exemple de méthode MUSCL satisfaisant toutes les inégalités d'entropie numériques, C.R. Acad. Sci. Paris, Sér. I. Math. 317 (1993), 619--624.MR 94e:65119
2: G. Chen and J. Liu, Convergence of second-order schemes for isentropic gas dynamics, Math. Comp. 61 (1993), 607-629. MR 94a:65047
3: B. Cockburn, F. Coquel, and F. LeFloch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), 77--103. MR 95d:65078
4: B. Cockburn and P. Gremaud, An error estimate for finite element methods for scalar conservation laws, SIAM J. Numer. Anal. (to appear).
5: F. Coquel and F. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp. 57 (1991), 169--210.MR 91m:65229
6: M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), 1--21. MR 81b:65079
7: R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27--70. MR 84k:35091
8: S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), 271--306. MR 22:10194
9: J. Goodman and R. LeVeque, On the accuracy of stable schemes for scalar conservation laws, Math. Comp. 45 (1985), 15--21. MR 86f:65149
10: 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 54:1640
11: G. Jennings, Discrete shocks, Comm. Pure Appl. Math. 27 (1974), 25--37. MR 49:3358
12: C. Johnson, A. Szepessy, and P. Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp. 54 (1990), 107--129.MR 90j:65118
13: S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 47 (1970), 217--243. MR 42:2159
14: 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--109.MR 58:3510
15: P. Lions and P. Souganidis, Convergence of MUSCL type methods for scalar conservation laws, C. R. Acad. Sci. Paris. Sér. I Math. 311 (1990), 259--264. MR 91i:65168
16: H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), 408--463. MR 91i:65157
17: S. Osher, Riemann solvers, the entropy condition and difference approximations, SIAM J. Numer. Anal. 21 (1984), 217--235. MR 86d:65119
18: ------, On convergence of generalized MUSCL schemes, SIAM J. Numer. Anal. 22 (1985), 947--961. MR 87b:65147
19: S. Osher and S. Chakravarthy, High resolution schemes and entropy condition, SIAM J. Numer. Anal. 21 (1984), 955--984. MR 86a:65086
20: S. Osher and E. Tadmor, On the convergence of difference schemes to scalar conservation laws, Math. Comp. 50 (1988), 19--51. MR 89m:65086
21: R. Sanders, On convergence of monotone finite difference schemes for variable spatial differencing, Math. Comp. 40 (1983), 19--36. MR 84a:65075
22: C.-W. Shu, Numerical solutions of conservation laws, Ph.D. thesis, UCLA, 1986.
23: G. Jiang and C.-W. Shu, On a cell entropy inequality for discontinuous Galerkin methods, Math. Comp. 62 (1994), 531--538. MR 94h:65099
24: P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984), 995--1011. MR 85m:65085
25: E. Tadmor, Convenient total variation diminishing conditions for nonlinear difference schemes, SIAM J. Numer. Anal. 25 (1988), 1002--1014. MR 90b:65173
26: A. Volpert, The space BV and quasi-linear equations, Math. USSR Sb. 73 (1967), 255--302. MR 35:7172
27: P. Woodward and P. Colella, The piecewise-parabolic method for gas-dynamical simulations, J. Comput. Phys. 54 (1984), 115--173. MR 85e:76004
28: H. Yang, An artificial compression method for ENO schemes: the slope modification method, J. Comput. Phys. 89 (1990), 125--160.MR 91g:65214
29: ------, Nonlinear wave analysis and convergence of MUSCL schemes, IMA Preprint Series 697, 1990.

Retrieve articles in Mathematics of Computation with MSC (1991): 65M60, 65M12, 35L65

Huanan Yang
Affiliation: Department of Mathematics Kansas State University Manhattan, Kansas 66506
Email: hyang@math.ksu.edu

DOI: 10.1090/S0025-5718-96-00668-0
PII: S 0025-5718(96)00668-0
Keywords: Conservation law, MUSCL schemes, schemes using flux limiters, entropy condition, convergence
Received by editor(s): December 20, 1993
Received by editor(s) in revised form: September 13, 1994 and January 30, 1995.
Copyright of article: Copyright 1996, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS