AN EXPLICIT MULTI-TIME STEPPING HYBRID SCHEME FOR THE ALLEN-CAHN EQUATION

초록

. In this paper, we develop an explicit hybrid multi-time-stepping finite difference method for the Allen-Cahn equation. The Allen-Cahn equation is a nonlinear parabolic phasefield model used for diffuse-interface and phase-transition problems. When the interfacial thickness parameter is small, a fine spatial mesh is required to resolve the transition layer, and standard explicit finite difference schemes suffer from severe time-step restrictions. These restrictions mainly come from the diffusive scale and the stiffness of the nonlinear reaction term. The proposed method is based on an operator splitting strategy. The Allen-Cahn equation is divided into a linear diffusion equation and a nonlinear reaction equation. For the diffusion equation, we use a recursive multi-time-stepping procedure. The first level is treated by a standard explicit diffusion step, and the subsequent levels are computed by DuFort-Frankel-type recursive updates. For the nonlinear reaction equation, we use the closed-form analytical solution at each grid point. Therefore, the method remains explicit in structure and does not require nonlinear iterations. Several numerical experiments are presented to examine the performance of the proposed method. The tests include an exact traveling-wave benchmark, discrete energy evolution, and curvature-driven interface motions in two- and three-dimensional space. The numerical results show that the proposed method captures interfacial dynamics well in the tested examples and exhibits the expected energy-decay behavior. The method can also use larger macro time-step sizes than the standard explicit hybrid method under the tested settings. These results suggest that the proposed scheme provides a practical explicit approach that balances accuracy, computational cost, and implementation simplicity for Allen-Cahn dynamics.

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