Linear energy stable and maximum principle preserving semi-implicit scheme for Allen–Cahn equation with double well potential

Abstract

In this paper, we consider numerical approximation of the Allen–Cahn equation with double well potential, which is a fundamental equation in phase-field models. We propose a novel linear, energy stable and maximum principle preserving scheme, which is obtained combining a recently developed energy factorization approach with a novel stabilization approach to treat the double well potential semi-implicitly. Different from the traditional stabilization approach, our stabilization approach aims to make sure the energy inequality in an enlarged phase variable domain. Compared with the prevalent convex-splitting approach and auxiliary variable approaches, the proposed approach leads to a very simple, linear scheme that preserves the original energy dissipation law. The proposed fully discrete finite difference scheme is proved to preserve the discrete maximum principle without any time stepping constraint. The performance of the proposed scheme is demonstrated in numerical experiments, and especially, it is vastly superior to the conventional stabilized scheme.