An innovative meshless approach for solving 2D Allen-Cahn equations using the RBF-compact finite difference method

This paper presents a numerical meshless approach for solving the two-dimensional Allen-Cahn equation, utilizing a radial basis function-compact finite difference (RBF-CFD) method in combination with the Strang splitting technique. The Allen-Cahn equation is crucial for modeling phase transitions and interface dynamics, making its numerical solution essential for applications in materials science, fluid dynamics, and biology. For spatial discretization, we employ the RBF-CFD method, which provides high-order accuracy. Specifically, the Hermite RBF interpolation technique is used to approximate the model operators over local stencils. For temporal discretization, we apply the Strang splitting method, which improves both accuracy and efficiency by decomposing the equation into simpler components. This approach is particularly effective for handling nonlinear equations and complex systems, ensuring the preservation of physical properties while minimizing numerical errors. To assess the method's performance, we conduct several numerical simulations to evaluate accuracy, stability, and convergence across different configurations. The results highlight the method's ability to maintain key qualitative properties, such as energy decay over time.