Unconditional energy stability and maximum principle preserving scheme for the Allen-Cahn equation

In this paper, we propose a novel fully implicit numerical scheme that satisfies both nonlinear energy stability and maximum principle for the space fractional Allen-Cahn equation. Especially, the fully implicit second-order scheme in time has never been proved to preserve the maximum principle before. For the resulting nonlinear scheme, we also propose a nonlinear iterative algorithm, which is uniquely solvable, convergent, and can preserve discrete maximum principle in each iterative step. Then we provide an error estimate by using the established maximum principle which plays a key role in the analysis. Several numerical experiments are presented to verify the theoretical results.