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

Abstract

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 α(1<α≤2). The proposed method is shown to be unconditionally stable in L2-norm, and satisfying the discrete maximum principle under some reasonable time step constraint. Finally, numerical experiments are given to verify our theoretical findings.