Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation

Abstract

In this study, we present the stability analysis of a fully explicit finite difference method (FDM) for solving the Allen–Cahn (AC) equation. The AC equation is a second-order nonlinear partial differential equation (PDE), which describes the antiphase boundaries of the binary phase separation. In the presented stability analysis, we consider the explicit Euler method for the temporal derivative and second-order finite difference in the space direction. The explicit scheme is fast and accurate because it uses a small time step, however, it has a temporal step constraint. We analyze and compute that the explicit time step constraint formula guarantees the discrete maximum principle for the numerical solutions of the AC equation. The numerical stability of the explicit scheme automatically holds when we use the time satisfying the discrete maximum principle. The computational numerical experiments demonstrate the stability, discrete maximum principle, and accuracy of the explicit scheme for the constrained time step. Furthermore, it is shown that the time step obtained is not severe restriction when we consider the temporal accuracy.