Investigations on several numerical methods for the non-local Allen-Cahn equation

This paper investigates some numerical methods for solving the non-local Allen-Cahn equation with a space-time dependent Lagrange multiplier. Several types of methods, including the Crank-Nicolson finite difference method, the finite difference operator splitting method, and the Fourier spectral operator splitting method are proposed respectively. Comparisons are made among these methods in terms of accuracy and computational efficiency for solving different types of problems. Numerical results show that the Fourier spectral operator splitting method is very efficient because of its spectral accuracy in space, especially for simulation of long time dynamics.