A fourth-order maximum principle preserving operator splitting scheme for three-dimensional fractional Allen-Cahn equations

Abstract: In this paper, by using Strang's second-order splitting method, the numerical procedure for the three-dimensional (3D) space fractional Allen-Cahn equation can be divided into three steps. The first and third steps involve an ordinary differential equation, which can be solved analytically. The intermediate step involves a 3D linear fractional diffusion equation, which is solved by the Crank-Nicolson alternating directional implicit (ADI) method. The ADI technique can convert the multidimensional problem into a series of one-dimensional problems, which greatly reduces the computational cost. A fourth-order difference scheme is adopted for discretization of the space fractional derivatives. Finally, Richardson extrapolation is exploited to increase the temporal accuracy. The proposed method is shown to be unconditionally stable by Fourier analysis. Another contribution of this paper is to show that the numerical solutions satisfy the discrete maximum principle under reasonable time step constraint. For fabricated smooth solutions, numerical results show that the proposed method is unconditionally stable and fourth-order accurate in both time and space variables. In addition, the discrete maximum principle is also numerically verified.