Preprint 1999-032

Operator Splitting Methods for Degenerate Convection-Diffusion Equations I: Convergence and Entropy Estimates

Helge Holden, Kenneth Hvistendahl Karlsen, and Knut-Andreas Lie

Abstract: We present and analyze a numerical method for the solution of a class of scalar, multi-dimensional, nonlinear degenerate convection-diffusion equations. The method is based on operator splitting to separate the convective and the diffusive terms in the governing equation. The nonlinear, convective part is solved using front tracking and dimensional splitting, while the nonlinear diffusion equation is solved by a suitable difference scheme. We verify $L^1$ compactness of the corresponding set of approximate solutions and derive precise entropy estimates. In particular, these results allow us to pass to the limit in our approximations and recover an entropy solution of the problem in question. The theory presented covers a large class of equations. Important subclasses are hyperbolic conservation laws, porous medium type equations, two-phase reservoir flow equations, and strongly degenerate equations coming from the recent theory of sedimentation-consolidation processes. A thorough numerical investigation of the method analyzed in this paper (and similar methods) is presented in a companion paper.

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Helge Holden, <>
Kenneth Hvistendahl Karlsen, <>
Knut-Andreas Lie, <>
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<> October 6 1999.

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