A Decompositional Approach for Two-Dimensional, Two-Phase, Nonlinear Inverse Stefan Problems Using the Method of Fundamental Solutions

In this paper, we propose a decompositional (phase-wise split) approach to solve a two-dimensional, two-phase (solid and liquid, say), nonlinear inverse Stefan problem. The first step is to approximate the unknown moving boundary between the two phases and the Stefan condition on that boundary using the overspecified boundary and initial data on the solid. The second and final step is then to reconstruct the temperature and heat flux on the fixed liquid boundary using the approximated Stefan conditions and given initial data. In each phase, we obtain the Tikhonov-regularized approximations using the method of fundamental solutions (MFS) and formulate heuristic residual a posteriori estimators to quantify the errors in the approximations. The MFS parameters for controlling the error are detected automatically in a systematic way, by virtue of a mean-filtering algorithm and a deterministic optimization strategy; this is in stark contrast to the less systematic way employed in existing nonlinear optimization algorithms. Numerical results demonstrate the effectiveness of the proposed approach.