A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems (Communications in Computational Physics, 2023)
Thermal phase change problems are widespread in mathematics, nature,and science. They are particularly useful in simulating the phenomena of meltingand solidification in materials science. In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phasechange model, which is the coupling of a heat transfer equation and a phase field equation. The unconditional energy stability and consistency error estimates are rigorouslyproved for the proposed schemes. A detailed implementation demonstrates that theproposed method requires only the solution of a system of linear elliptic equations ateach time step, with an efficient scheme of sufficient accuracy to calculate the solutionat the first step. It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps. Adaptive time stepsize strategies can be applied to further benefit from this unconditional stability. Numerical experiments are presented to verify the theoretical claims and to illustrate theaccuracy and effectiveness of our method.