We present a simple and practical adaptive finite difference method for the Allen–Cahn (AC) equation in the two-dimensional space. We use a temporally adaptive narrow band domain embedded in the uniform discrete rectangular domain. The narrow band domain is defined as a neighboring region of the interface. We employ a recently developed explicit hybrid numerical scheme for the AC equation. Therefore, the computational algorithm on the narrow band discrete domain is simple and fast. We demonstrate the high performance of the proposed adaptive method for the AC equation through various computational experiments.

中文翻译：

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Allen-Cahn方程的实用自适应网格方法

我们为二维空间中的Allen-Cahn（AC）方程提供了一种简单实用的自适应有限差分方法。我们使用嵌入到统一离散矩形域中的时间自适应窄带域。窄带域定义为接口的相邻区域。我们为AC方程采用了最近开发的显式混合数值方案。因此，在窄带离散域上的计算算法既简单又快速。我们通过各种计算实验证明了所提出的交流方程自适应方法的高性能。