Convergence for long-times of a semidiscrete Perona-Malik equation in one dimension

We rigorously prove that the semidiscrete schemes of a Perona-Malik type equation con-verge, in a long time scale, to a suitable system of ordinary differential equations defined on piecewise constant functions. The proof is based on a formalasymptotic expansion argument, and on a careful construction of discrete comparison functions. Despite the equation has a region where it is backward parabolic, we prove a discrete comparison principle, which is the key tool for the convergence result.