Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient

Abstract

We analyze approximate solutions generated by 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 γ(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 Δ , which is a manifestation of resonance. To circumvent this analytical prob- lem, we construct a singular mapping Ψ(γ,·) such that the total variation of the transformed variable z Δ =Ψ ( γ Δ ,u Δ ) can be bounded uniformly in Δ. This establishes strong L 1 com- pactness of z Δ and, since Ψ(γ,·) is invertible, also u Δ . 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 Kruzkov-type entropy inequality. We prove that the diffusion function A(u )i s Hcontin- uous, 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.