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

We analyse 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 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(Delta), which is a manifestation of resonance. To circumvent this analytical problem, we construct a singular mapping Psi(gamma, .) such that the total variation of the transformed variable z(Delta) = Psi(gamma(Delta), u(Delta)) can be bounded uniformly in Delta. This establishes strong L-1 compactness of z(Delta) and, since Psi(gamma, .) is invertible, also u(Delta). 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) is Holder 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.