Analysis of Numerical Solution of the Stefan Problem

A Stefan problem is a problem involving a parabolic differential equation with a moving boundary. We study one particular one-dimensional, one phase Stefan problem and two numerical methods for solving it. The first method which has been published by J. Douglas and T. M. Gallie is a finite difference method with variable step size in the t-direction. We supply a convergence proof for the iteration which, at each time step, is needed to determine the size of the step. We also derive certain estimates which we use subsequently to obtain bounds for the solution functions of the original problem. We also discuss stability of the method showing partial results but without being able to prove stability.We prove that the boundary curve of the particular Stefan Problem in question is monotone increasing with a derivative that tends to zero as t tends to infinity. Furthermore, we show that the temperature at the t-axis, u(0,t), goes asymptoticly like 2.sqrt(t/ ¼)