An explicit hybrid finite difference scheme for the Allen-Cahn equation

In this paper, we propose an explicit hybrid numerical method for solving the Allen-Cahn equation, which models antiphase domain coarsening process in a binary mixture. The proposed method is based on an operator splitting method. First, we solve the linear diffusion part using the explicit Euler method. Second, we solve the nonlinear term using the closed-form analytical solution. We show the stability condition of the proposed numerical scheme. We also show the pointwise boundedness of the numerical solution for the Allen-Cahn equation under a solvability condition. Numerical experiments such as linear stability analysis, traveling wave, motion by mean curvature, image segmentation, and crystal growth are presented to demonstrate the performance of the proposed method.