A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation

In this paper, a numerical method is proposed to solve the space fractional Allen-Cahn equation. Based on Crank-Nicolson method for time discretization and second-order weighted and shifted Grunwald difference formula for spatial discretization, we present a new linearized two-level scheme, where the nonlinear term is handled by Newton linearized technology. And we only need to solve a linear system at each time level. Then, the unique solvability of the numerical scheme is given. Under the appropriate assumptions of time step, the discrete maximum principle and energy stability of the numerical scheme are proved. Furthermore, we give a detailed error analysis, which reflects that the temporal and spatial convergence orders are both second order. At last, some numerical experiments show that the proposed method is reasonable and effective.