Convergence of a semidiscrete scheme for a forward-backward parabolic equation

We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation ut = (W ′(ux))x with periodic boundary conditions in one space dimension, where W is a standard double-well potential. We characterize the equation satisfied by the limit of the discretized solutions as the grid size goes to zero. Using an approximation argument, we show that it is possible to flow initial data u having regions where ux falls within the concave region {W ′′ < 0} of W , where the backward character of the equation manifests. It turns out that the limit equation depends on the way we approximate u in its unstable region.