Degenerate parabolic equation with zero flux boundary condition and its approximations

TL;DR

This paper investigates a degenerate parabolic-hyperbolic equation with zero flux boundary conditions, proving convergence of a finite volume scheme to the unique entropy solution using new analytical tools and solution concepts.

Contribution

It introduces a new notion of integral-process solution and establishes convergence of an implicit finite volume scheme for zero flux boundary problems.

Findings

Convergence of numerical solutions to the entropy solution.

Development of new analytical tools for zero flux boundary conditions.

Establishment of fundamental estimates for the scheme.

Abstract

We study a degenerate parabolic-hyperbolic equation with zero flux boundary condition. The aim of this paper is to prove convergence of numerical approximate solutions towards the unique entropy solution. We propose an implicit finite volume scheme on admissible mesh. We establish fundamental estimates and prove that the approximate solution converge towards an entropy-process solution. Contrarily to the case of Dirichlet conditions, in zero-flux problem unnatural boundary regularity of the flux is required to establish that entropy-process solution is the unique entropy solution. In the study of well-posedness of the problem, tools of nonlinear semigroup theory (stationary, mild and integral solutions) were used in [Andreianov, Gazibo, ZAMP, 2013] in order to overcome this difficulty. Indeed, in some situations including the one-dimensional setting, solutions of the stationary problem enjoy additional boundary regularity. Here, similar arguments are developed based on the new notion of integral-process solution that we introduce for this purpose.