In this article, a two‐step discretization method based on multi‐quadrics (MQ) radial basis function (RBF) is presented for solving Allen–Cahn (AC) equation with integer derivative for time and space. In the first step, backward Euler formula with Newton iterative method is used to discrete the time direction of AC equation. And RBF method is applied in space for solving semi‐discrete linearized problem on a coarse mesh. In the second step, finite difference (FD) and radial basis function‐finite difference (RBF‐FD) methods are used to solve the problem on a fine mesh, respectively. Numerical tests for the equation are obtained to verify the feasibility and computational efficiency of the considered process. In addition, the comparison between FD and RBF‐FD shows that solutions obtained by RBF‐FD are higher accuracy.

中文翻译：

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基于RBF-FD方法的2D / 3D Allen-Cahn方程的两步离散化方法

本文提出了一种基于多二次方（MQ）径向基函数（RBF）的两步离散化方法，用于求解时空整数导数的Allen-Cahn（AC）方程。第一步，采用牛顿迭代法的反向欧拉公式来离散交流方程的时间方向。为了解决粗糙网格上的半离散线性化问题，在空间中应用了RBF方法。第二步，分别使用有限差分（FD）和径向基函数-有限差分（RBF-FD）方法解决细网格上的问题。对该方程进行了数值测试，以验证所考虑过程的可行性和计算效率。此外，FD与RBF‐FD的比较表明，RBF‐FD所获得的解具有更高的精度。