Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

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1. [1] M. Afif and B. Amaziane, Convergence of finite volume schemes for a degenerate convection-diffusion equation arising in two-phase flow in porous media. Preprint (1999). Zbl1052.65516MR2062160
2. [2] F. Bouchut, F.R. Guarguaglini and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations. Indiana Univ. Math. J. 49 (2000) 723–749. Zbl0964.35011
3. [3] R. Bürger, S. Evje and K.H. Karlsen, On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl. 247 (2000) 517–556. Zbl0961.35078
4. [4] M.C. Bustos, F. Concha, R. Bürger and E.M. Tory, Sedimentation and thickening: Phenomenological foundation and mathematical theory. Kluwer Academic Publishers, Dordrecht (1999). Zbl0936.76001MR1747460
5. [5] J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999) 269–361. Zbl0935.35056
6. [6] C. Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate. RAIRO-Modél. Math. Anal. Numér. 33 (1999) 129–156. Zbl0921.65071
7. [7] S. Champier, T. Gallouët and R. Herbin, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math. 66 (1993) 139–157. Zbl0801.65089
8. [8] B. Cockburn, F. Coquel and P. Le Floch, An error estimate for finite volume methods for multidimensional conservation laws. Math. Comp. 63 (1994) 77–103. Zbl0855.65103
9. [9] B. Cockburn, F. Coquel and P.G. LeFloch, Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32 (1995) 687–705. Zbl0845.65051
10. [10] B. Cockburn and P.-A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. I. The general approach. Math. Comp. 65 (1996) 533–573. Zbl0848.65067
11. [11] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463 (electronic). Zbl0927.65118
12. [12] M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980) 1–21. Zbl0423.65052
13. [13] M.G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78 (1980) 385–390. Zbl0449.47059
14. [14] B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321–351. Zbl0469.65067
15. [15] M.S. Espedal and K.H. Karlsen, Numerical solution of reservoir flow models based on large time step operator splitting algorithms, in Filtration in Porous media and industrial applications. Lect. Notes Math. 1734, Springer, Berlin (2000) 9–77. Zbl1077.76546
16. [16] S. Evje and K.H. Karlsen, Discrete approximations of $BV$ solutions to doubly nonlinear degenerate parabolic equations. Numer. Math. 86 (2000) 377–417. Zbl0963.65094
17. [17] S. Evje and K.H. Karlsen, Degenerate convection-diffusion equations and implicit monotone difference schemes, in Hyperbolic problems: Theory, numerics, applications, Vol. I (Zürich, 1998). Birkhäuser, Basel (1999) 285–294. Zbl0931.65094
18. [18] S. Evje and K.H. Karlsen, Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations. Numer. Math. 83 (1999) 107–137. Zbl0961.65084
19. [19] S. Evje and K.H. Karlsen, Monotone difference approximations of $BV$ solutions to degenerate convection-diffusion equations. SIAM J. Numer. Anal. 37 (2000) 1838–1860 (electronic). Zbl0985.65100
20. [20] S. Evje and K.H. Karlsen, Second order difference schemes for degenerate convection-diffusion equations. Preprint (in preparation). Zbl0931.65094
21. [21] R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18 (1998) 563–594. Zbl0973.65078
22. [22] R. Eymard, T. Gallouët, D. Hilhorst and Y. Naït Slimane, Finite volumes and nonlinear diffusion equations. RAIRO-Modél. Math. Anal. Numér. 32 (1998) 747–761. Zbl0914.65101
23. [23] 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) 635–648. Zbl0776.35034
24. [24] A. Harten, J.M. Hyman and P.D. Lax, On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. XXIX (1976) 297–322. Zbl0351.76070
25. [25] H. Holden, K.H. Karlsen and K.-A. Lie, Operator splitting methods for degenerate convection-diffusion equations I: Convergence and entropy estimates, in Stochastic processes, physics and geometry: New interplays. A volume in honor of Sergio Albeverio. Amer. Math. Soc. (to appear). Zbl0974.35065MR1803424
26. [26] H. Holden, K.H. Karlsen, K.-A. Lie and N.H. Risebro, Operator splitting for nonlinear partial differential equations: An $L^1$ convergence theory. Preprint (in preparation). Zbl0989.65093
27. [27] E. Isaacson and B. Temple, Convergence of the $2\times 2$ Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625–640. Zbl0838.35075
28. [28] K.H. Karlsen and N.H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Preprint, Department of Mathematics, University of Bergen (2000). Zbl1027.35057MR1974417
29. [29] C. Klingenberg and N.H. Risebro, Stability of a resonant system of conservation laws modeling polymer flow with gravitation. J. Differential Equations March (2000). Zbl0977.35083MR1815188
30. [30] 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. Zbl0836.35090
31. [31] D. Kröner, S. Noelle and M. Rokyta, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numer. Math. 71 (1995) 527–560. Zbl0841.65079
32. [32] D. Kröner and M. Rokyta, Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions. SIAM J. Numer. Anal. 31 (1994) 324–343. Zbl0856.65104
33. [33] S.N. Kružkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof. Mat. Zametki 6 (1969) 97–108. Zbl0189.10602
34. [34] S.N. Kružkov, First order quasi-linear equations in several independent variables. Math. USSR Sbornik 10 (1970) 217–243. 
35. [35] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241–282. Zbl0987.65085
36. [36] N.N. Kuznetsov, Accuracy of some approximative methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comput. Math. Math. Phys. Dokl. 16 (1976) 105–119. Zbl0381.35015
37. [37] B.J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque. SIAM J. Numer. Anal. 22 (1985) 1074–1081. Zbl0584.65059
38. [38] S. Noelle, Convergence of higher order finite volume schemes on irregular grids. Adv. Comput. Math. 3 (1995) 197–218. Zbl0834.65088
39. [39] M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. Preprint, Mathematische Fakultät, Albert-Ludwigs-Universität Freiburg (2000). MR1825703
40. [40] O.A. Oleĭnik, Discontinuous solutions of non-linear differential equations. Amer. Math. Soc Transl. Ser. 2 26 (1963) 95–172. Zbl0131.31803
41. [41] S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19–51. Zbl0637.65091
42. [42] É. Rouvre and G. Gagneux, Solution forte entropique de lois scalaires hyperboliques-paraboliques dégénérées. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 599–602. Zbl0935.35085
43. [43] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-up in quasilinear parabolic equations. Walter de Gruyter & Co., Berlin (1995). Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors. Zbl1020.35001MR1330922
44. [44] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing. Math. Comp. 40 (1983) 91–106. Zbl0533.65061
45. [45] B. Temple, Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws. Adv. in Appl. Math. 3 (1982) 335–375. Zbl0508.76107
46. [46] J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. Preprint, Available at the URL http://www.math.ntnu.no/conservation/ Zbl0972.65060MR1770068
47. [47] J. Towers, A difference scheme for conservation laws with a discontinuous flux - the nonconvex case. Preprint, Available at the URL http://www.math.ntnu.no/conservation/ Zbl1055.65104MR1870839
48. [48] J.-P. Vila, Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes. RAIRO-Modél. Math. Anal. Numér. 28 (1994) 267–295. Zbl0823.65087
49. [49] A.I. Vol’pert, The spaces BV and quasi-linear equations. Math. USSR Sbornik 2 (1967) 225–267. Zbl0168.07402
50. [50] A.I. Vol’pert and S.I. Hudjaev, Cauchy’s problem for degenerate second order quasilinear parabolic equations. Math. USSR Sbornik 7 (1969) 365–387. Zbl0191.11603