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 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 L 1 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.1991 Mathematics Subject Classification. 35K65, 35L45, 35L65, 65M06, 65M12.

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

“…Strongly degenerate parabolic equations with discontinuous fluxes are studied in [21,22,53]. In [21] equations like (1.2) are studied with a concave convective flux u → f (γ(x), u) and with (γ 1 (x)A(u) x ) x replaced by A(u) xx .…”

“…Note that our pointwise discretization of γ, (3.2), follows the usage of [16,20,21,22], but differs from that of [19], where γ is discretized by cell averages taken over the cells [x j , x j+1 ), where x j := j∆x, j ∈ Z. The important point is that in both cases, the discretization of γ is staggered with respect to that of the conserved quantity u, and this property greatly facilitates the convergence analysis of the numerical schemes.…”

“…This equation is discretized in space by a first-order conservative finite volume scheme using the Engquist-Osher approximation, for which convergence results for our class of problems are available [16,19,20,21,22]. For time discretization an explicit Euler scheme is used.…”

Operating Charts for Continuous Sedimentation II: Step Responses

“…Some further recent references on the modelling and simulation of the entire clarificationthickening process (with varying theoretical support) are [34][35][36][37][38][39][40][41][42]. In particular, the important contribution by Bürger et al [42], which relies on the analyses by Karlsen et al [43,44], contains a generalization of the previous results for the hyperbolic equation to the case when also compression at high concentrations is modelled, which leads to a hyperbolic-parabolic partial differential equation.…”

Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations

“…This was first exploited by Evje and Karlsen in [1]. Related analyses include implicit monotone schemes for degenerate parabolic equations [14], problems with boundary conditions [15], multidimensional degenerate parabolic equations [16], equations with discontinuous coefficients [17,18,19], and problems of parameter identification [20] (this list is far from being complete). Of course, the robustness of monotone schemes, in particular the convergence to the entropy solution, comes at the well-known price of the generic limitation to first-order accuracy.…”