A Stochastic method for determination of the phase change boundary

The determination of solid/liquid interfaces implies the solution of the Stefan problem, which involves either one heat equation in an unknown domain (one phase Stefan problem) or two heat equations in unknown domains. The determination of the domain(s) is a core difficulty of the solution, for which many approaches may be found in the literature. Among a large number of them, we may cite those based on variational inequalities, regularization, surface measures, shape optimization and signed measures. In design applications, it is necessary to make repeated calls to the solver of Stefan problem, which may generate a high computational cost. To alleviate the cost, an alternative is the use of stochastic diffusions and Feyman-Kac representations of the solutions of heta equations. In these approaches, the unknown is the boundary between the solid and liquid regions, which is characterized as the solution of an algebraic equation. Thus, it may be determined by algebraic numerical methods. Since the equation involves the field of temperatures, it is necessary to determine it, at least on the regions near the boundary: in this step, Feyman-Kac representations and stochastic diffusion are useful, since they furnish the result with a reasonable accuracy and rapidly, what saves computational effort. We present here a method for the determination of the free boundary of Stefan's problem, with numerical examples that show that the approach is effective to calculate. Comparisons with a Finite Element approach are presented.