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

Area of Science:

• Computational physics and materials science
• Numerical analysis and scientific computing

Background:

• The Allen-Cahn equation is fundamental for simulating phase transitions and interface dynamics in various scientific fields.
• Accurate and efficient numerical solutions are critical for applications in materials science, fluid dynamics, and biology.

Purpose of the Study:

• To present a novel numerical meshless approach for solving the two-dimensional Allen-Cahn equation.
• To enhance the accuracy and efficiency of simulating phase transitions and interface dynamics.

Main Methods:

• Utilized a radial basis function-compact finite difference (RBF-CFD) method for spatial discretization, employing Hermite RBF interpolation for high-order accuracy.
• Implemented the Strang splitting technique for temporal discretization to improve accuracy and efficiency by decomposing the equation.
• Combined RBF-CFD with Strang splitting for a robust numerical solution of nonlinear equations.

Main Results:

• Numerical simulations demonstrated the method's high-order accuracy, stability, and convergence.
• The approach effectively preserved key qualitative properties, including energy decay over time.
• Validated the method's performance across various configurations for complex systems.

Conclusions:

• The proposed RBF-CFD and Strang splitting method offers an accurate and efficient approach for solving the 2D Allen-Cahn equation.
• This numerical technique is well-suited for modeling phase transitions and interface dynamics in scientific and engineering applications.
• The method's ability to maintain physical properties ensures reliable simulation results.