Analysis of the operator splitting scheme for the Allen-Cahn equation

This paper presents two numerical schemes for solving the Allen-Cahn equation representing a model for antiphase domain coarsening in a binary mixture. Based on the operator splitting method, the Allen-Cahn equation was divided into a linear and a nonlinear sub equation: the linear sub equation was solved by a Fourier spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size; the nonlinear sub equation was then solved analytically due to the availability of a closed-form solution. The stability and error analysis of the numerical solution are presented in detail. Numerical experiments are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, we show that the schemes are unconditionally stable and second-order accurate in time.