An unconditionally stable hybrid numerical method for solving the Allen Cahn equation

We present an unconditionally stable second-order hybrid numerical method for solving the Allen Cahn equation representing a model for antiphase domain coarsening in a binary mixture. The proposed method is based on operator splitting techniques. The Allen Cahn equation was divided into a linear and a nonlinear equation. First, the linear equation was discretized using a Crank Nicolson scheme and the resulting discrete system of equations was solved by a fast solver such as a multigrid method. The nonlinear equation was then solved analytically due to the availability of a closed-form solution. Various numerical experiments are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, we show that the scheme is unconditionally stable and second-order accurate in both time and space.