On the maximum principle and energy stability for fully discretized fractional-in-space Allen-Cahn equation

We consider numerical methods for solving the fractional-in-space Allen-Cahn (FiSAC) equation which contains small perturbation parameters and strong noninearilty. Standard fully discretized schemes for the the FiSAC equation will be considered, namely, in time the conventional first-order implicit-explicit scheme or second-order Crank-Nicolson scheme and in space a secondorder finite difference approach. The main purpose of this work is to establish discrete stability in both maximum and energy norms. In particular, we will show that the numerical results obtained by using the fully discretized schemes are conditionally stable: both the discrete maximum principle and the energy decaying properties are preserved under certain restrictions on the time step. Moreover, our analysis applied to one to three space dimensions. Numerical experiments are performed to verify the theoretical results.