A meshless method combined with operator splitting for solving the Allen-Cahn equation

초록

We propose a meshless computational method that combines Strang operator splitting with stabilization methodologies to preserve the maximum principle of the Allen-Cahn (AC) equation. The AC equation remains an active research topic because of the ongoing need for high-performance numerical methods that preserve essential characteristics such as the maximum principle, energy dissipation, and unconditional stability. The proposed method adopts a hybrid framework that uses multiquadric (MQ) interpolation for spatial discretization and the finite difference method for temporal integration. Stabilization methods are incorporated to rigorously establish the maximum principle. The performance of the proposed method is validated through computational tests. An important advantage of this mesh-free approach is its use of MQ interpolation at each time step, which facilitates straightforward implementation. Furthermore, numerical results for cases involving scattered data points demonstrate the superiority of the proposed method.

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