Numerical Study on an RBF-FD Tangent Plane Based Method for Convection-Diffusion Equations on Anisotropic Evolving...

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Differential Equations

Background:

• Solving partial differential equations (PDEs) on evolving surfaces is crucial for modeling dynamic phenomena.
• Existing methods often face challenges with accuracy and stability on complex geometries.
• Convection-diffusion equations are fundamental in describing transport processes in various scientific fields.

Purpose of the Study:

• To develop a robust and accurate numerical method for solving convection-diffusion PDEs on evolving surfaces.
• To enhance the RBF-FD method for handling anisotropic surface growth.
• To improve the stability and accuracy of numerical solutions for dynamic surface problems.

Main Methods:

• A fully Lagrangian method based on the radial basis function finite difference (RBF-FD) approach.
• Discretization of surface differential operators using the tangent plane approximation with Gaussian RBFs and 2D polynomials.
• Coupling the RBF-FD method with anisotropic RBFs (ARBFs) for improved accuracy in anisotropic growth scenarios.
• Modification of the RBF interpolation metric to match anisotropic surface geometry.

Main Results:

• The proposed method simplifies the calculation of differentiation weights.
• Demonstrated improved accuracy and stability for convection-diffusion equations on evolving surfaces.
• Successfully applied to problems involving anisotropic surface growth and coupled PDE solutions.
• ARBF interpolation effectively handles anisotropic surface geometries.

Conclusions:

• The developed fully Lagrangian RBF-FD method offers a stable and accurate approach for PDEs on evolving surfaces.
• The integration of ARBFs significantly enhances the capability to model anisotropic surface dynamics.
• This method provides a valuable tool for simulating complex physical processes on dynamic geometries.