An Efficient High-Order Meshless Method for Advection-Diffusion Equations on Time-Varying Irregular Domains

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Scientific Computing

Background:

• Solving advection-diffusion equations on time-varying domains presents significant computational challenges.
• Existing methods often require parameter tuning or struggle with efficiency on moving grids.

Purpose of the Study:

• To develop a high-order, parameter-free computational framework for advection-diffusion equations on time-varying domains.
• To enable efficient and accurate simulations on complex, deforming geometries.

Main Methods:

• A generalized Overlapped Radial Basis Function Finite Difference (RBF-FD) method with automatic weight computation.
• A novel procedure for tuning-free assembly and updating of differentiation matrices on moving domains.
• Integration with a high-order semi-Lagrangian method and time-integration for advection-diffusion.

Main Results:

• Demonstrated high-order convergence for advection-diffusion on time-varying 2D and 3D domains across various Peclet numbers.
• Verified O(N log N) time complexity, confirming computational efficiency.
• Successfully applied the framework to a 3D problem modeling platelet aggregation and coagulation.

Conclusions:

• The developed RBF-FD framework provides a robust, efficient, and parameter-free solution for advection-diffusion on moving domains.
• The method's ability to handle complex geometries and achieve high-order accuracy makes it suitable for challenging scientific applications.