An unconditionally stable fast high order method for thermal phase change models

Abstract

Thermal phase change problems arise in a large number of applications. In this paper, we consider a phase field model instead of the classical Stefan model to describe phenomena, which may appear in some complex phase change problems such as dendritic crystal growth, phase transformations in metallic alloys, etc. Our aim is to propose efficient and accurate schemes for the model, which is the coupling of a heat transfer equation and a phase field equation. The schemes are constructed based on an auxiliary variable approach for the phase field equation and semi-implicit treatment for the heat transfer equation. The main novelty of the paper consists in: (i) construction of the efficient schemes, which only requires solving several second-order elliptic problems with constant coefficients; (ii) proof of the unconditional stability of the schemes; (iii) fast high order solver for the resulting equations at each time step. A series of numerical examples are presented to verify the theoretical claims and to illustrate the efficiency of our method. As far as we know, it seems this is the first attempt made for the thermal phase change model of this type.