Abstract
We analyse implicit monotone finite difference schemes for nonlinear, possibly strongly degenerate, convection-diffusion equations in one spatial dimension. Since we allow strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data. We thus choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. This paper complements our previous work [8] on explicit monotone schemes.
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References
H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial value problem for u t − ΔØ(u) = 0, J. Math. Pures et Appl, 58 (1979), 153–163.
R. Bürger and W. L. Wendland, Existence, uniqueness and stability of generalized solutions of an initial-boundary value problem for a degenerating parabolic equation, J. Math. Anal. AppL, 218 (1998), 207–239.
B. Cockburn and G. Gripenberg, Continuous Dependence on the Nonlinearities of Solutions of Degenerate Parabolic Equations, Preprint, IMA, University of Minnesota, Minneapolis, (1997).
F. Concha, and R. Bürger, Wave propagation phenomena in the theory of sedimentation. In: Toro, E.F. and Clarke, J.F. (eds.), Numerical Methods for Wave Propagation, Kluwer Academic Publishers, Dordrecht, (May, 1998).
M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1–21.
M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general anach spaces, Amer. J. Math., 93 (1971), 265–298.
S. Evje and K. H. Karlsen, Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations., Submitted to Numer. Math., Preprint, Institut Mittag-Leffler, Stockholm, (1997).
S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations, Submitted to Siam J. Num. Anal., Preprint, University of Bergen, (1998).
S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik, 10 (1970), 217–243.
S. N. Kruzkov, Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications, Mat. Zametki, 6 (1969), 97–108.
B. J. Lucier, On non-local monotone difference schemes for scalar conservation laws, Math. Comp., 47 (1986), 19–36.
A. I. Volpert and S. I. Hudjaev, Cauchy’s problem for degenerate second order quasi-linear parabolic equations, Math. USSR Sbornik, 7 (1969), 365–387.
Z. Wu and J. Yin, Some properties of functions in BV and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J, 5 (1989), 395–422.
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© 1999 Springer Basel AG
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Evje, S., Karlsen, K.H. (1999). Degenerate Convection-Diffusion Equations and Implicit Monotone Difference Schemes. In: Fey, M., Jeltsch, R. (eds) Hyperbolic Problems: Theory, Numerics, Applications. International Series of Numerical Mathematics, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8720-5_31
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DOI: https://doi.org/10.1007/978-3-0348-8720-5_31
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9742-6
Online ISBN: 978-3-0348-8720-5
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