Linear relaxation schemes for the Allen-Cahn-type and Cahn-Hilliard-type phase field models

Abstract

This letter introduces novel linear relaxation schemes for solving the phase field models, particularly the Allen-Cahn (AC) type and Cahn-Hilliard (CH) type equations. The proposed schemes differ from existing schemes for the phase field models in the literature. The resulting semi-discrete schemes are linear by discretizing the AC and CH models on staggered time meshes. Only a linear algebra problem needs to be solved at each time marching step after the spatial discretization. Furthermore, our proposed schemes are shown to be unconditionally energy stable, i.e., the numerical solutions respect energy dissipation laws without restriction on the time steps. Several numerical examples are provided to illustrate the power of the proposed linear relaxation schemes for solving phase field models.