An explicit stable finite difference method for the Allen–Cahn equation

Abstract

We propose an explicit stable finite difference method (FDM) for the Allen–Cahn (AC) equation. The AC equation has been widely used for modeling various phenomena such as mean curvature flow, image processing, crystal growth, interfacial dynamics in material science, and so on. For practical use, an explicit method can be applied for the numerical approximation of the AC equation. However, there is a strict restriction on the time step size. To mitigate the disadvantage, we adopt the alternating direction explicit method for the diffusion term of the AC equation. As a result, we can use a relatively larger time step size than when the explicit method is used. Numerical experiments are performed to demonstrate that the proposed scheme preserves the intrinsic properties of the AC equation and it is stable compared to the explicit method.