In this article, we will explore the numerical simulation of the Allen–Cahn equation and provide effective combination methods to efficiently solve it. The Allen–Cahn equation, an equation of mathematical physics, represents a singularly perturbed reaction–diffusion phenomenon that elucidates the phase separation mechanism occurring in multi-component alloy systems. Finding a numerical solution for the Allen–Cahn equation presents a difficult challenge for computational science and engineering researchers. To solve the Allen–Cahn equation, we will use Lie–Trotter’s and Strang’s splitting techniques combined with the radial basis function partition of unity method. We will discretize the problem’s spatial domain using the classical and direct RBF-PU methods. We will also provide suitable theorems for analytical support of the presented numerical methods. We will provide several numerical examples which include examples with exact solutions to check the accuracy and efficiency of the method and examples in the field of phase transition to demonstrate the adaptability, performance, and efficiency of the proposed method.

中文翻译：

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Allen-Cahn问题的二算子分裂局部径向基函数方法的数值研究

在本文中，我们将探索 Allen–Cahn 方程的数值模拟，并提供有效的组合方法来高效求解该方程。艾伦-卡恩方程是一个数学物理方程，代表一种奇异微扰反应扩散现象，阐明了多组分合金系统中发生的相分离机制。寻找艾伦-卡恩方程的数值解对计算科学和工程研究人员来说是一项艰巨的挑战。为了求解 Allen–Cahn 方程，我们将使用 Lie–Trotter 和 Strang 分裂技术并结合统一方法的径向基函数划分。我们将使用经典的直接 RBF-PU 方法离散化问题的空间域。我们还将提供合适的定理来支持所提出的数值方法的分析。我们将提供几个数值示例，其中包括具有精确解的示例以检查该方法的准确性和效率，以及相变领域的示例以证明该方法的适应性、性能和效率。