The Numerical Analysis of a Phase Field Model in Moving Boundary Problems

Abstract

Moving boundary problems arise in a large number of applications including the phenomena of melting and solidification. The classical treatment is the Stefan model which ignores some of the important factors such as surface tension, supercooling, and superheating during the process. A new model called “the phase field model,” which captures such phenomena in pure and isotropic materials, is considered. It contains as a special case the classical Stefan model in the sense that there is diffusion away from the free surface and liberation of latent heat at phase transitions. Based on a Helmholtz free energy functional, a phase function indicating the phase state of the material is coupled with emperature to form a phase field model which contains a set of parabolic type equations. By using the theory of positively invariant regions, a global existence theorem is given. Numerical schemes are suggested along with stability and convergence studies. In one dimension the results of computation are compared with the analytic solution and show good agreement. Numerical simulations of the solidification process under supercooling in two dimensions are also given. These results clearly show the conflict between the destabilizing force of supercooling and the stabilizing force of surface tension.