Numerical Analysis of Crank-Nicolson Scheme for the Allen-Cahn Equation

Abstract

We consider numerical methods to solve the Allen-Cahn equation using the second-order Crank-Nicolson scheme in time and the second-order central difference approach in space. The existence of the finite difference solution is proved with the help of Browder fixed point theorem. The difference scheme is showed to be unconditionally convergent in $L_∞$ norm by constructing an auxiliary Lipschitz continuous function. Based on this result, it is demonstrated that the difference scheme preserves the maximum principle without any restrictions on spatial step size and temporal step size. The numerical experiments also verify the reliability of the method.