Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation

Abstract

In this paper, we will investigate the first- andsecond-order implicit-explicit schemeswith parameters for solving the Allen-Cahn equation.It is known that the Allen-Cahn equation satisfies a nonlinearstability property, i.e., the free-energy functional decreasesin time. The goalof this paper is to consider implicit-explicit schemes that inheritthe nonlinear stability of the continuous model, which will beachieved by properly choosing parameters associated with theimplicit-explicit schemes. Theoretical justifications forthe nonlinear stability of the schemes will beprovided, and the theoretical results will be verified byseveral numerical examples.