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We study a degenerate parabolic-hyperbolic equation with zero flux boundarycondition. The aim of this paper is to prove convergence of numericalapproximate solutions towards the unique entropy solution. We propose animplicit finite volume scheme on admissible mesh. We establish fundamentalestimates and prove that the approximate solution converge towards anentropy-process solution. Contrarily to the case of Dirichlet conditions, inzero-flux problem unnatural boundary regularity of the flux is required toestablish that entropy-process solution is the unique entropy solution. In thestudy 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 situationsincluding the one-dimensional setting, solutions of the stationary problemenjoy additional boundary regularity. Here, similar arguments are developedbased on the new notion of integral-process solution that we introduce for thispurpose.

Degenerate parabolic equation with zero flux boundary condition and its approximations