A fourth-order finite difference method for the Allen-Cahn equation

In this study, we present a spatially fourth-order accurate hybrid numerical scheme for the Allen-Cahn (AC) equation in two-dimensional (2D) and three-dimensional (3D) spaces. The proposed hybrid numerical method splits the AC model into nonlinear and linear components using the operator splitting technique. The nonlinear component is solved by using an analytic solution. In 3D space, the linear diffusion term is solved by splitting it into the x-, y-, and z-directional single spatial variable diffusion equations. The fully implicit scheme for temporal difference and the spatially fourth-order finite difference discretization are applied. The system of discrete equations becomes a penta-diagonal matrix that can be directly solved without any iterative techniques. Stability analysis and various computational experiments are performed to verify the numerical convergence and stability of the proposed method in 2D and 3D spaces. Furthermore, we compared the convergence rate, error, and CPU time between the proposed fourth-order and standard second-order schemes.