On the interface identification of free boundary problem by method of fundamental solution

Abstract

SUMMARYThe Stefan problems with one phase have been studied thoroughly. For the numerical implementations of free‐boundary problems, the unknown free boundary makes the mesh generation process much difficult. The purpose of this paper is to give a meshless numerical scheme on the basis of the method of fundamental solution for solving a two‐phase Stefan problem with variable diffusion coefficients. By introducing some mathematical transform and artificial boundary, this problem is solved approximately by constructing an explicit solution with the superposition of finite number of source points, in terms of the fundamental solution to a standard heat equation. The superposition coefficients of solutions as well as the free boundary are solved by a least square solution for a nonlinear algebra system in terms of the initial and boundary value conditions. Our proposed scheme can be considered as a generalization of numerical linear algebra system applied to the PDEs by method of fundamental solution. Numerical examples are presented to show the validity of the proposed scheme. Copyright © 2012 John Wiley & Sons, Ltd.