Fast and Accurate Numerical Approaches for Stefan Problems and Crystal Growth

In this paper, some new approaches are proposed for moving boundary/interface problems, particularly for the Stefan problems and the problem of unstable crystal growth. We will focus on the issues of accuracy and speed-up. A modiied Crank-Nicolson method which is second order accurate and stable is developed. The ADI (alternating implicit directional method) is also developed to speed up the simulation for certain class of problems. The ADI method is shown to be asymptotic stable and at least rst order accurate. Numerical results, however, showed that the ADI method actually is second order accurate if the velocity can be calculate accurately. The level set method is used to update the moving interface so that the topological changes can be handled easily. Numerical experiments are presented against an exact solution and those results appeared in the literature.