CONVERGENCE FOR LONG-TIMES OF A SEMIDISCRETE PERONA–MALIK EQUATION IN ONE DIMENSION

Abstract

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