Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients

Mathematical Models and Methods in Applied Sciences

Volumn 13, Issue 2, 2003, Pages 221-257

  Seguin, Nicolas ^a,b   Vovelle, Julien ^a  

^a CNRS   (France)

^b EDF R AND D   (France)

Author keywords

Conservation law; Discontinuous coefficient; Finite volume; Resonance

Indexed keywords

EID: 0037295910     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202503002477     Document Type: Article

References (35)

1. T. Buffard, T. Gallouët and J.-M. Hérard, A sequel to a rough Godunov scheme: Application to real gases, Comput. Fluids 29 (2000) 813-847.
2. F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal. 32 (1998) 891-933.
3. C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer-Verlag, 2000).
4. S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal. 28 (1995) 1425-1451.
5. G. Dal Maso, P. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483-548.
6. R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis (North-Holland, 2000), Vol. VII, pp. 713-1020.
7. T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Comput. Fluids 32 (2003) 479-513.
8. T. Gallouët, J.-M. Hérard and N. Seguin, Some recent finite volume method to compute Euler equations using real gas EOS, Int. J. Numer. Methods Fluids 39 (2002) 1073-1138.
9. T. Gallouët, J.-M. Hérard and N. Seguin, Numerical modelling of two-phase flows using the two-fluid two-pressure approach, EDF-DRD Report HI-81/01/043/A (2001).
10. E. Giusti, Minimal Surfaces and Functions of Bounded Variation (Birkhäuser, 1984).
11. J. M. Greenberg and A. Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33 (1996) 1-16.
12. S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. 47 (1959) 271-306.
13. T. Gimse and N. H. Risebro, Riemann problems with a discontinuous flux function, in Third Int. Conf. Hyperbolic Problems (Uppsala, 1990), Studentlitteratur, Lund, 1991, Vols. I, II, pp. 488-502.
14. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws (Springer-Verlag, 1996).
15. E. Godlewski and P.-A. Raviart, The numerical coupling of nonlinear hyperbolic systems of conservation laws: I. The scalar case, preprint, 2001.
16. A. Harten and J. M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys. 50 (1983) 235-269.
17. E. Isaacson, D. Marchesin, B. Plohr and B. Temple, The Riemann problem near a hyperbolic singularity: The classification of solutions of quadratic Riemann problems. I, SIAM J. Appl. Math. 48 (1988) 1009-1032.
18. E. Isaacson and B. Temple, Analysis of a singular hyperbolic system of conservation laws, J. Differential Equations 65 (1986) 250-268.
19. E. Isaacson and B. Temple, The Riemann problem near a hyperbolic singularity. II, III, SIAM J. Appl. Math. 48 (1988) 1287-1301; 1302-1318.
20. E. Isaacson and B. Temple, Nonlinear resonance in inhomogeneous systems of conservation laws, in Mathematics of Nonlinear Science (Phoenix, AZ, 1989) (Amer. Math. Soc., 1990), pp. 63-77.
21. R. A. Klausen and N. H. Risebro, Stability of conservation laws with discontinuous coefficients, J. Differential Equations 157 (1999) 41-60.
22. C. Klingenberg and N. H. Risebro, Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations 20 (1995) 1959-1990.
23. C. Klingenberg and N. H. Risebro, Stability of a resonant system of conservation laws modeling polymer flow with gravitation, J. Differential Equations 170 (2001) 344-380.
24. K. H. Karlsen, N. H. Risebro and J. D. Towers, On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient, No. 13.
25. S. N. Kružkov, First-order quasilinear equations with several independent variables, Mat. Sb. 81 (1970) 228-255.
26. A. Y. Le Roux, Riemann solvers for some hyperbolic problems with a source term, Actes du 30ème Congèrs d'Analyse Numérique: CANum '98 (Arles, 1998) (Soc. Math. Appl. Indust., 1999), pp. 75-90 (electronic).
27. L. Lin, J. B. Temple and J. Wang, A comparison of convergence rates for Godunov's method and Glimm's method in resonant nonlinear systems of conservation laws, SIAM J. Numer. Anal. 32 (1995) 824-840.
28. L. W. Lin, B. Temple and J. H. Wang, Suppression of oscillations in Godunov's method for a resonant non-strictly hyperbolic system, SIAM J. Numer. Anal. 32 (1995) 841-864.
29. N. Seguin, Génération et validation de Rozavel, un code équilibre en hydraulique 2D, mémoire de D.E.A., Université de Bordeaux I, June 1999, in French.
30. B. Temple, Global solution of the Cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws, Adv. Appl. Math. 3 (1982) 335-375.
31. J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal. 38 (2000) 681-698 (electronic).
32. J. D. Towers, A difference scheme for conservation laws with a discontinuous flux: The nonconvex case, SIAM J. Numer. Anal. 39 (2001) 1197-1218 (electronic).
33. A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Rational Mech. Anal. 160 (2001) 181-193.
34. B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys. 32 (1979) 101-136.
35. A. I. Vol'pert, Spaces bv and quasilinear equations, Mat. Sb. 73 (1967) 255-302.