Free access
Issue
ESAIM: M2AN
Volume 35, Number 2, March/April 2001
Page(s) 239 - 269
DOI http://dx.doi.org/10.1051/m2an:2001114
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).
  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]
  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]
  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).
  5. J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999) 269-361. [CrossRef] [MathSciNet]
  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.
  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]
  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]
  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]
  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]
  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]
  12. M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980) 1-21. [CrossRef] [MathSciNet]
  13. M.G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78 (1980) 385-390. [MathSciNet]
  14. B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321-351. [CrossRef] [MathSciNet]
  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.
  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]
  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.
  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]
  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]
  20. S. Evje and K.H. Karlsen, Second order difference schemes for degenerate convection-diffusion equations. Preprint (in preparation).
  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]
  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]
  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]
  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.
  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).
  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).
  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]
  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).
  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).
  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. [CrossRef] [MathSciNet]
  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]
  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. [CrossRef] [MathSciNet]
  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]
  34. S.N. Kruzkov, First order quasi-linear equations in several independent variables. Math. USSR Sbornik 10 (1970) 217-243. [CrossRef]
  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. [NASA ADS] [CrossRef] [MathSciNet]
  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. [CrossRef]
  37. B.J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque. SIAM J. Numer. Anal. 22 (1985) 1074-1081. [CrossRef] [MathSciNet]
  38. S. Noelle, Convergence of higher order finite volume schemes on irregular grids. Adv. Comput. Math. 3 (1995) 197-218. [CrossRef] [MathSciNet]
  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).
  40. O.A. Oleĭnik, Discontinuous solutions of non-linear differential equations. Amer. Math. Soc Transl. Ser. 2 26 (1963) 95-172.
  41. S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51. [CrossRef] [MathSciNet]
  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.
  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.
  44. R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing. Math. Comp. 40 (1983) 91-106. [CrossRef] [MathSciNet]
  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]
  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/
  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/
  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. [MathSciNet]
  49. A.I. Vol'pert, The spaces BV and quasi-linear equations. Math. USSR Sbornik 2 (1967) 225-267. [CrossRef]
  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]

Recommended for you