Convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function

SIAM Journal on Numerical Analysis

Volumn 43, Issue 2, 2005, Pages 559-577

  Mishra, Siddhartha ^a,b  

^a INDIAN INSTITUTE OF SCIENCE   (India)

^b TATA INSTITUTE OF FUNDAMENTAL RESEARCH   (India)

Author keywords

Conservation laws; Discontinuous fluxes; Finite differences; Singular mapping

Indexed keywords

CONFORMAL MAPPING; CONVERGENCE OF NUMERICAL METHODS; FUNCTION EVALUATION; MATHEMATICAL OPERATORS; NUMERICAL METHODS;

CONSERVATION LAWS; DISCONTINUOUS FLUXES; SINGULAR MAPPING;

FINITE DIFFERENCE METHOD;

EID: 33646482469     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/030602745     Document Type: Article

References (29)

1. ADIMURTHI AND G. D. V. GOWDA, Conservation laws with discontinuous flux, J. Math. Kyoto Univ., 43 (2003), pp. 27-70.
2. ADIMURTHI AND G. D. V. GOWDA, Extensions of Godunov Scheme for Conservation Laws with Flux Function Discontinuous in the Space Variable, preprint, 2003.
3. ADIMURTHI, J. JAFFRÉ, AND G. D. V. GOWDA, Codunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal., 42 (2004), pp. 179-208.
4. ADIMURTHI, S. MISHRA, AND G. D. V. GOWDA, Conservation laws with flux function discontinuous in the space variable. I. Optimal entropy solutions, J. Hyperbolic Differ. Equ., submitted.
5. ADIMURTHI, S. MISHRA, AND G. D. V. GOWDA, Conservation laws with flux function discontinuous in the space variable. II. Convex-concave type fluxes and generalised entropy solutions, J. Comput. Appl. Math., submitted.
6. ADIMURTHI, S. MISHRA, AND G. D. V. GOWDA, Convergence of Godunov Type Methods for Conservation Laws with a Spatially Varying Discontinuous Flux Function, preprint.
7. E. AUDUSSE AND B. PERTHAME, Uniqueness for a Scalar Conservation Law with Discontinuous Flux via Adapted Entropies, INRIA report-5261, 2004.
8. P. BAGNERINI AND M. RASCLE, A multiclass homogenized hyperbolic model of traffic flow, SIAM J. Math. Anal., 35 (2003), pp. 949-973.
9. R. BURGER, K. H. KARLSEN, N. H. RISEBRO, AND J. D. TOWERS, Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math., 97 (2004), pp. 25-65.
10. G. M. COCLITE AND N. H. RISEBRO, Conservation Laws with Time Dependent Discontinuous Coefficients, preprint.
11. S. DIEHL, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26 (1995), pp. 1425-1451.
12. S. DIEHL, A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Math., 56 (1996), pp. 388-419.
13. B. ENQUIST AND S. OSHER, One sided difference approximations for nonlinear conservation laws, Math. Comp., 34 (1980), pp. 321-351.
14. T. GIMSE AND N. H. RISEBRO, Riemann problems with discontinuous flux function, in Proceedings of the 3rd Annual International Conference on Hyperbolic Problems, Studentlitteratur, Uppsala, 1991, pp. 488-502.
15. T. GIMSE AND N. H. RISEBRO, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), pp. 635-648.
16. E. GODLEWSKI AND P. A. RAVIART, Hyperbolic Systems of Conservation Laws, Math. Appl. (Paris), Ellipses, Paris, 1991.
17. K. H. KARLSEN AND J. D. TOWERS, Convergence of the Lax-Friedrichs scheme and stability of conservation laws with a discontinuous space-time dependent flux, Chinese Ann. Math. Ser B., 25 (2004), pp. 287-318.
18. K. H. KARLSEN, C. KLINGENBERG, AND N. H. RISEBRO, A relaxation scheme for conservation laws with a discontinuous coefficients, Math. Comp., 73 (2004), pp. 1235-1259.
19. K. H. KARLSEN, N. H. RISEBRO, AND J. D. TOWERS, Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient, IMA J. Numer. Anal., 22 (2002), pp. 623-664.
20. K. H. KARLSEN, N. H. RISEBRO, AND J. D. TOWERS, On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient, Electron. J. Differential Equations, 2002 (93) (2002), 23 pp. (electronic).
21. 1 stability for entropy solution of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk., no. 3, (2003), pp. 1-49.
22. R. A. KLAUSEN AND N. H. RISEBRO, Stability of conservation laws with discontinuous coefficients, J. Differential Equations, 157 (1999), pp. 41-60.
23. C. KLINGENBERG AND N. H. RISEBRO, Convex conservation laws with discontinuous coefficients, existence, uniqueness and asymptotic behaviour, Comm. Partial Differential Equations, 20 (1995), pp. 1959-1990.
24. C. KLINGENBERG AND N. H. RISEBRO, Stability of a resonant system of conservation laws modeling polymer flow with gravitation, J. Differential Equations, 170 (2001), pp. 344-380.
25. S. N. KRUZKHOV, First order quasilinear in several independent variables, Math. USSR Sb., 10 (1970), pp. 217-243.
26. S. MISHRA, Scalar Conservation Laws with Discontinuous Flux, Master's thesis, Indian Institute of Science, Bangalore, India, 2003.
27. B. TEMPLE, Global solution of the Cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), pp. 335-375.
28. J. D. TOWERS, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), pp. 681-698.
29. J. D. TOWERS, A difference scheme for conservation laws with a discontinuous flux: The nonconvex case, SIAM J. Numer. Anal., 39 (2001), pp. 1197-1218.