Comparison study of numerical methods for solving the Allen-Cahn equation

The goal of this paper is to present a brief review and a critical comparison of the performance of several numerical schemes for solving the Allen-Cahn equation representing a model for antiphase domain coarsening in a binary mixture. Explicit, fully implicit, Crank-Nicolson, and unconditionally gradient stable schemes are considered. In this paper, we show the solvability conditions of the numerical schemes and the decreasing property of total energy using eigenvalues of the Hessian matrix of the energy functional. We also present the pointwise boundedness of the numerical solution for the Allen-Cahn equation. To compare the accuracy and numerical efficiency of these methods, numerical experiments such as traveling wave and motion by mean curvature are performed. Numerical results show that Crank-Nicolson and nonlinearly stabilized splitting schemes are almost close to the analytic solution. However, if a large time step is used in the numerical test, we have only two results with linearly and nonlinearly stabilized splitting schemes in spite of having large gaps between analytic solution and numerical results. The other numerical schemes except for linearly and nonlinearly stabilized splitting schemes have unstable results when large time step is used.