Stability of the totally implicit finite difference scheme for the one- dimensional diffusion equation with a moving boundary (Q2639622)

The totally implicit \(O(h^ 2k)\) finite difference method applied to a linear one-dimensional diffusion equation with a boundary moving with constant velocity is analyzed. The related physical problem is the solidification of a finite column filled with a binary mixture, in a quasiequilibrium regime. By advancing the boundary along the path h/k \(=\) velocity of the interface, a simple algorithm is obtained which is shown to be consistent and unconditionally stable for any value of the interfacial segregation factor and of the Peclet number.