  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.
