An Application of the Method of Lines to Multidimensional Free Boundary Problems

Abstract

Multidimensional free boundary problems for elliptic and parabolic differential equations, among them one phase Stefan problems, are partially discretized with the method of lines. The multi-point free boundary problem for the resulting system of second order ordinary differential equations is solved iteratively with an SOR type iteration. At each step of this iteration a free boundary problem for a single ordinary differential equation must be solved for which an initial value technique is used. The method can effectively cope with quite general space and time dependent free surface conditions.