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

In this paper, we will investigate the first- and second-order implicitexplicit schemes with parameters for solving the Allen-Cahn equation. It is known that the Allen-Cahn equation satisfies a nonlinear stability property, i.e., the free-energy functional decreases in time. The goal of this paper is to consider implicit-explicit schemes that inherit the nonlinear stability of the continuous model, which will be achieved by properly choosing parameters associated with the implicit-explicit schemes. Theoretical justifications for the nonlinear stability of the schemes will be provided, and the theoretical results will be verified by several numerical examples.