A maximum bound principle preserving CN/AB finite difference scheme for Riesz space-fractional Allen-Cahn equations with logarithmic free energy

It is well known that the Allen-Cahn equation satisfies the maximum bound principle and the energy dissipation law. Such two properties are important in the study of the stability of the solution to the Allen-Cahn equation, and whether they could be inherited at the discrete level is a significant issue in numerical simulations. In this paper, we consider finite difference approximation of the Riesz space-fractional Allen-Cahn equation with small perturbation parameter and logarithmic free energy. The proposed scheme is obtained based on the combination of a second-order Crank-Nicolson/Adams-Bashforth (CN/AB) scheme for temporal approximation and a second-order finite difference approach for spatial discretization. A stabilized term is added to the discretized scheme to maintain numerical stability. It is shown that the numerical solutions satisfy the discrete maximum bound principle under reasonable constraints on the time step size and the stabilization parameter. Based on the maximum-norm stability, the discrete energy stability and $L^{\infty}$ error estimate are investigated. Some numerical experiments are performed to verify the theoretical results.