A maximum-principle preserving and unconditionally energy-stable linear second-order finite difference scheme for Allen–Cahn equations

Abstract

In this paper, we propose a new linear second-order finite difference scheme for Allen–Cahn equations. We use a modified Leap-Frog finite difference scheme with stabilized term and a central finite difference scheme for temporal and spatial discretization respectively. It is shown that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Based on the maximum stability, the maximum-norm error is analyzed. Moreover, we can see that the proposed scheme is unconditionally energy-stable. Finally, a numerical experiment is performed to verify the theoretical results.