Weakly nonoscillatory schemes for scalar conservation laws

Mathematics of Computation

Volumn 72, Issue 244, 2003, Pages 1747-1767

  Kopotun, Kirill ^a   Neamtu, Marian ^b   Popov, Bojan ^c  

^a UNIVERSITY OF MANITOBA   (Canada)

^b VANDERBILT UNIVERSITY   (United States)

^c TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Error estimates; Godunov type schemes; Relaxed entropie projection; Scalar conservation laws; Weak nonoscillation

Indexed keywords

EID: 0141976338     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01524-2     Document Type: Article

References (23)

1. F. Bouchut, Ch. Bourdarias and B. Perthame, A MUSCL method satisfying all entropy inequalities, Math. Comp. 65 (1996), 1439-1461. MR 97a:65080
2. F. Bouchut and B. Perthame, Kružkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2847-2870. MR 98m:65156
3. Y. Brenier and S. Osher, The one-sided Lipschitz condition for convex scalar conservation laws, SIAM J. Numer. Anal. 25 (1988), 8-23. MR 89a:65134
4. P. Colella and P. Woodward, The piecewise parabolic method for gas-dynamical simulations, J. Comput. Phys. 54 (1984), 174-201.
5. F. Coquel and P. LeFloch, An entropy satisfying MUSCL scheme for systems of conservation laws, Numer. Math. 74 (1996), 1-33. MR 97g:65179
6. R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993. MR 95f:41001
7. A. Harten and S. Osher, Uniformly high order accurate nonoscillatory schemes, I, SIAM J. Numer. Anal. 24 (1987), no. 2, 279-309. MR 90a:65198
8. A. Harten, B. Enquist, S. Osher and S.R. Chakravarthy, Uniformly high order accurate essentially nonoscillatory schemes, III, J. Comput. Phys. 71 (1987), no. 2, 231-303. MR 90a:65199
9. G.-S. Jiang, C.-T. Lin, S. Osher and E. Tadmor High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM J. Numer. Anal. 35 (1998), no. 6, 2147-2169. MR 99j:65145
10. G.-S. Jiang and E. Tadmor Nonoscillatory central schemes for hyperbolic conservation laws, SIAM J. Sci. Comput. 19 (1998), no. 6, 1892-1917. MR 99f:65128
11. S.N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik 10 (1970), no. 2, 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 equations, USSR Comput. Math, and Math. Phys. 16 (1976), no. 6, 105-119. MR 58:3510
13. P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217-237. MR 22:11523
14. X. Liu and S. Osher, Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes, I, SIAM J. Numer. Anal. 33 (1996), 760-779. MR 97h:65110
15. X. Liu and E. Tadmor, Third order nonoscillatory central scheme for hyperbolic conservation laws, Numer. Math. 79 (1997), 397-425. MR 99h:65149
16. B. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), no. 6, 1074-1081. MR 88a:65104
17. H. Nessyahu and E. Tadmor, The convergence rate of nonlinear scalar conservation laws, SIAM J. Numer. Anal. 29 (1992), 1505-1519. MR 93j:65139
18. S. Osher and E. Tadmor, On the convergence rate of difference approximations to scalar conservation laws, Math. Comp. 50 (1988), 19-51. MR 89m:65086
19. R. Rockafellar and R. Wets, Variational analysis, Springer-Verlag, Berlin, 1998. MR 98m:49001
20. C.-W. Shu, Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Sci. Comput. 5 (1990), no. 2, 127-149.
21. E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), 891-906. MR 92d:35190
22. _, Approximate solutions of Nonlinear Conservation Laws and Related Equations, Proc. Sympos. Appl. Math., 54 (1998), 325-368. MR 99c:35149
23. W. Ziemer, Weakly Diflerentiable Functions, Graduate Texts in Mathematics, Springer-Verlag, 1989. MR 91e:46046