On the preserving of the maximum principle and energy stability of high-order implicit-explicit Runge-Kutta schemes for the space-fractional Allen-Cahn equation

Abstract

We put forward and analyze the high-order (up to fourth) strong stability-preserving implicit-explicit Runge-Kutta schemes for the time integration of the space-fractional Allen-Cahn equation, which inherits the maximum principle preserving and energy stability. The space-fractional Allen-Cahn equation with homogeneous Dirichlet boundary condition is first discretized in the spatial direction by using a second-order fractional centered difference scheme that preserves the semi-discrete maximum principle. It is subsequently integrated in the temporal direction by a class of strong stability-preserving implicit-explicit Runge-Kutta schemes that are specifically designed to preserve the maximum principle to the optimal time step size. The convergence order in the discrete $L^{\infty }$ norm and energy boundedness are provided by using the established maximum principle. Finally, a series of numerical experiments are carried out to demonstrate the high-order convergence, maximum principle preserving, and energy stability of the proposed schemes.