Fast evolution numerical method for the Allen–Cahn equation

We present a fast evolution numerical algorithm for solving the Allen–Cahn (AC) equations. One of efficient computational techniques for the AC equation is the operator splitting method. We split the AC equation into the linear heat and nonlinear equations; and then solve the linear part using the Fourier spectral method and the nonlinear part using an analytic closed-form solution. These steps are unconditionally stable. However, if a large time step is used, then the nonlinear part dominates the evolution and results in a sharp interfacial transition layer. To overcome these problems, we propose a time rescaling method to the nonlinear part of the AC equation. Computational tests verify the performance of the proposed method which makes the evolution fast and interfacial transition layer be uniform.