Numerical Analysis of Crank-Nicolson Scheme for the Allen-Cahn Equation (Journal of Computational Mathematics, 2021)
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 inspace. The existence of the finite difference solution is proved with the help of Browderfixed 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 withoutany restrictions on spatial step size and temporal step size. The numerical experimentsalso verify the reliability of the method.