On wavewise entropy inequalities for high-resolution schemes. I: The semidiscrete case

Mathematics of Computation

Volumn 65, Issue 213, 1996, Pages 45-67

  Yang, Huanan ^a  

^a KANSAS STATE UNIVERSITY   (United States)

Author keywords

Conservation law; Convergence; Entropy condition; MUSCL schemes; Schemes using flux limiters

Indexed keywords

CAUCHY PROBLEM; CONSERVATION LAW; FLUX LIMITATION; HIGH RESOLUTION; HYPERBOLIC EQUATION; WAVEWISE ENTROPY INEQUALITY;

CONVERGENCE OF NUMERICAL METHODS; NUMERICAL ANALYSIS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 0030353899     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-96-00668-0     Document Type: Article

References (29)

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 2D 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. Kružkov, 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. Acacl. 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. Vol'pert, 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 (PPM) 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 097, 1990.