First-order L1-convergence for relaxation approximations to conservation laws

Communications on Pure and Applied Mathematics

Volumn 51, Issue 8, 1998, Pages 857-895

  Teng, Zhen Huan ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032361628     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(sici)1097-0312(199808)51:8<857::aid-cpa1>3.0.co;2-4     Document Type: Article

References (26)

1. Bouchut, F., and Perthame, B., Kružhkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., to appear.
2. Caflisch, R. E., Jin, S., and Russo, G., Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Numer. Anal. 34, 1997, pp. 246-281.
3. Chen, G. Q, Levermore, C. D., and Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47, 1994, pp. 787-830.
4. Crandall, M. G., and Majda, A., Monotone difference approximations for scalar conservation laws, Math. Comp. 34, 1980, pp. 1-21.
5. Dafermos, C. M., Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26, 1977, pp. 1097-1119.
6. Goodman, J., and Xin, Z. P., Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rational Mech. Anal. 121, 1992, pp. 235-265.
7. Jin, S., and Levermore, C. D., Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 126, 1996, pp. 449-467.
8. Jin, S., Pareschi, L., and Toscani, G., Diffusive relaxation schemes for discrete-velocity kinetic equations, preprint, 1997.
9. Jin, S., and Xin, Z. P., The relaxation schemes for systems of conservation laws in arbitraryspace dimensions, Comm. Pure Appl. Math. 48, 1995, pp. 235-276.
10. Katsoulakis, M. A., and Tzavaras, A. E., Contractive relaxation systems and the scalar multidimensional conservation law. Comm. Partial Differential Equations 22, 1997, pp. 195-233.
11. Kružkov, S. N., First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81, 1970, pp. 228-255.
12. Kurganov, A., and Tadmor, E., Stiff systems of hyperbolic conservation laws: convergence and error estimates, SIAM J. Math. Anal. 28, 1997, pp. 1446-1456.
13. Kuznetsov, N. N., 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, pp. 105-119.
14. Lax, P. D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10, 1957, pp. 537-566.
15. Li, B. H., and Wang, J. H., The global qualitative study of solutions of a conservation law, Sci. Sinica 1979, Special Issue I on Math., pp. 12-38.
16. Liu, T.-P., Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108, 1987, pp. 153-175.
17. Natalini, R., Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49, 1996, pp. 795-823.
18. Oleǐnik, O. A., Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26, 1963, pp. 95-172.
19. Şabac, F., The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws, SIAM J. Numer. Anal. 34, 1997, pp. 2306-2318.
20. Schaeffer, D. G., A regularity theorem for conservation laws, Advances in Math. 11, 1973, pp. 368-386.
21. Tadmor, E., and Tassa, T., On the piecewise smoothness of entropy solutions to scalar conservation laws, Comm. Partial Differential Equations 18, 1993, pp. 1631-1652.
22. 1-error estimate for monotone difference schemes, Math. Comp. 64, 1995, pp. 581-589.
23. Tang, T., and Teng, Z.-H., Viscosity methods for piecewise smooth solutions to scalar conservation laws. Math. Comp. 66, 1997, pp. 495-526.
24. 1-rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks, SIAM J. Numer. Anal. 34, 1997, pp. 959-978.
25. Tveito, A., and Winther, R., On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term, SIAM J. Math. Anal. 28, 1997, pp. 136-161.
26. 1 convergence rate of viscous solutions of genuinely nonlinear scalar conservation laws, SIAM J. Math. Anal., to appear.