Convergence of a semidiscrete scheme for a forward-backward parabolic equation
The authors investigate the convergence of a semi-discrete scheme for the forward-backward parabolic equation \(u_t=(W'(u_x))_x\) with periodic boundary conditions in one space dimension, where \(W\) is a standard double-well potential. The authors characterize the equation satisfied by the limit of the discretized solutions as the grid size goes to zero. By an approximation argument, they prove that it is possible to flow initial data \(\overline{u}\) having regions where \(\overline{u}\) falls within the concave region \({W''<0}\) of \(W,\) where the backward character of the equation manifests itself. It is proved that the limit equation depends on the way one approximates \(\overline{u}\) in its unstable region.
scientific article; zbMATH DE number 6166939