A spatial fourth-order maximum principle preserving operator splitting scheme for the multi-dimensional fractional Allen-Cahn equation

In this paper, we consider the numerical study for the multi-dimensional fractional-in-space Allen-Cahn equation with homogeneous Dirichlet boundary condition. By utilizing Strang's second-order splitting method, at each time step, the numerical scheme can be split into three sub-steps. The first and third sub-steps give the same ordinary differential equation, where the solutions can be obtained explicitly. While a multi-dimensional linear fractional diffusion equation needs to be solved in the second sub-step, and this is computed by the Crank-Nicolson scheme together with alternating directional implicit (ADI) method. Thus, instead of solving a multidimensional nonlinear problem directly, only a series of one-dimensional linear problems need to be solved, which greatly reduces the computational cost. A fourth-order quasi-compact difference scheme is adopted for the discretization of the space Riesz fractional derivative of alpha(1 < alpha <= 2). The proposed method is shown to be unconditionally stable in L-2-norm, and satisfying the discrete maximum principle under some reasonable time step constraint. Finally, numerical experiments are given to verify our theoretical findings. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.