Maximum norm error analysis of an unconditionally stable semi-implicit scheme for multi-dimensional Allen-Cahn equations

In this paper, a linearized finite difference scheme is proposed for solving the multi-dimensional Allen-Cahn equation. In the scheme, a modified leap-frog scheme is used for the time discretization, the nonlinear term is treated in a semi-implicit way, and the central difference scheme is used for the discretization in space. The proposed method satisfies the discrete energy decay property and is unconditionally stable. Moreover, a maximum norm error analysis is carried out in a rigorous way to show that the method is second-order accurate both in time and space variables. Finally, numerical tests for both two- and three-dimensional problems are provided to confirm our theoretical findings.