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In this paper, by using Strang's second-order splitting method, the numericalprocedure for the three-dimensional (3D) space fractional Allen-Cahn equationcan be divided into three steps. The first and third steps involve an ordinarydifferential equation, which can be solved analytically. The intermediate stepinvolves a 3D linear fractional diffusion equation, which is solved by theCrank-Nicolson alternating directional implicit (ADI) method. The ADI techniquecan convert the multidimensional problem into a series of one-dimensionalproblems, which greatly reduces the computational cost. A fourth-orderdifference scheme is adopted for discretization of the space fractionalderivatives. Finally, Richardson extrapolation is exploited to increase thetemporal accuracy. The proposed method is shown to be unconditionally stable byFourier analysis. Another contribution of this paper is to show that thenumerical solutions satisfy the discrete maximum principle under reasonabletime step constraint. For fabricated smooth solutions, numerical results showthat the proposed method is unconditionally stable and fourth-order accurate inboth time and space variables. In addition, the discrete maximum principle isalso numerically verified.

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