Numerical study of two operator splitting localized radial basis function method for Allen–Cahn problem

Abstract

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.