Convergence of An Upwind Difference Scheme for Degenerate Parabolic Convection-Diffusion Equations with A Discontinuous Coefficient

Kenneth H. Karlsen, Nils H. Risebro, and John D. Towers

Abstract: We establish convergence of an upwind difference scheme (of Engquist-Osher type) for nonlinear degenerate parabolic convection-diffusion equations where the nonlinear convective flux function has a discontinuous coefficient $\gamma(x)$ and the diffusion function $A(u)$ is allowed to be strongly degenerate (the pure hyperbolic case is included in our setup). The main problem is obtaining a uniform bound on the total variation of the difference approximation $u^\D$, which is a manifestation of resonance. To circumvent this analytical problem, we construct a singular mapping $\Psi(\gamma,\cdot)$ such that the total variation of the transformed variable $z^\D=\Psi(\gamma^\D,u^\D)$ can be bounded uniformly in $\D$. This establishes strong $L^1$ compactness of $z^\D$ and, since $\Psi(\gamma,\cdot)$ is invertible, also $u^\D$. Our singular mapping is novel in that it incorporates a contribution from the diffusion function $A(u)$. We then show that the limit of a converging sequence of difference approximations is a weak solution as well as satisfying a Kru\v{z}kov-type entropy inequality. We prove that the diffusion function $A(u)$ is H\"older continuous, implying that the constructed weak solution $u$ is continuous in those regions where the diffusion is nondegenerate. Finally, some numerical experiments are presented and discussed.

Paper:
Available as PDF (355 Kbytes) or gzipped PDF (216 Kbytes; uncompress using gunzip).
Author(s):
Kenneth H. Karlsen , <kennethk@mi.uib.no>
Nils H. Risebro, <nilshr@math.uio.no>
John D. Towers, <jtowers@cts.com>
Publishing information:
UCLA Computational and Applied Mathematics Report