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 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 problem, 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 L¹ compactness 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 Kružkov‐type entropy inequality. We prove that the diffusion function A(u) is Hölder 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.