Free Access
Volume 35, Number 2, March/April 2001
Page(s) 239 - 269
Published online 15 April 2002
  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). [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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). [Google Scholar]
  5. J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999) 269-361. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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). [CrossRef] [MathSciNet] [Google Scholar]
  12. M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980) 1-21. [Google Scholar]
  13. M.G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78 (1980) 385-390. [MathSciNet] [Google Scholar]
  14. B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321-351. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  16. S. Evje and K.H. Karlsen, Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations. Numer. Math. 86 (2000) 377-417. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  18. S. Evje and K.H. Karlsen, Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations. Numer. Math. 83 (1999) 107-137. [CrossRef] [MathSciNet] [Google Scholar]
  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). [CrossRef] [MathSciNet] [Google Scholar]
  20. S. Evje and K.H. Karlsen, Second order difference schemes for degenerate convection-diffusion equations. Preprint (in preparation). [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  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). [Google Scholar]
  26. H. Holden, K.H. Karlsen, K.-A. Lie and N.H. Risebro, Operator splitting for nonlinear partial differential equations: An L1 convergence theory. Preprint (in preparation). [Google Scholar]
  27. E. Isaacson and B. Temple, Convergence of the 2 x 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625-640. [CrossRef] [MathSciNet] [Google Scholar]
  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). [Google Scholar]
  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). [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  33. S.N. Kruzkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof. Mat. Zametki 6 (1969) 97-108. [MathSciNet] [Google Scholar]
  34. S.N. Kruzkov, First order quasi-linear equations in several independent variables. Math. USSR Sbornik 10 (1970) 217-243. [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  37. B.J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque. SIAM J. Numer. Anal. 22 (1985) 1074-1081. [CrossRef] [MathSciNet] [Google Scholar]
  38. S. Noelle, Convergence of higher order finite volume schemes on irregular grids. Adv. Comput. Math. 3 (1995) 197-218. [CrossRef] [MathSciNet] [Google Scholar]
  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). [Google Scholar]
  40. O.A. Oleĭnik, Discontinuous solutions of non-linear differential equations. Amer. Math. Soc Transl. Ser. 2 26 (1963) 95-172. [Google Scholar]
  41. S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51. [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  44. R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing. Math. Comp. 40 (1983) 91-106. [Google Scholar]
  45. B. Temple, Global solution of the Cauchy problem for a class of 2 x 2 nonstrictly hyperbolic conservation laws. Adv. in Appl. Math. 3 (1982) 335-375. [CrossRef] [MathSciNet] [Google Scholar]
  46. J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. Preprint, Available at the URL [Google Scholar]
  47. J. Towers, A difference scheme for conservation laws with a discontinuous flux - the nonconvex case. Preprint, Available at the URL [Google Scholar]
  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. [Google Scholar]
  49. A.I. Vol'pert, The spaces BV and quasi-linear equations. Math. USSR Sbornik 2 (1967) 225-267. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]

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