Hybrid numerical method for the Allen–Cahn equation on nonuniform grids

In this article, we present a hybrid numerical scheme for solving the Allen–Cahn (AC) equation on a nonuniform mesh. The AC equation represents a model for antiphase domain coarsening in a binary mixture. To solve the AC equation on nonuniform grids, the AC equation is split into linear and nonlinear terms applying the operator splitting method. As the first step, the nonlinear term is solved using the separation of variables. Next, the diffusion term is decomposed into linear operators in each space direction. In each direction, we sequentially solve each diffusion equation by applying the implicit Euler method on a nonuniform grid to update the numerical solution. Because the implicit Euler method and analytic solution do not depend on the time step size, the proposed hybrid numerical method for the AC equation on a nonuniform mesh is unconditionally stable. In addition, we prove the proposed scheme satisfies the maximum principle. To verify the superior performance of the proposed method, we conduct numerical simulations such as motion by mean curvature, total energy non-increasing property, the maximum principle and unconditional stability.