  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.
