Hyperviscosity-Based Stabilization for Radial Basis Function-Finite Difference (RBF-FD) Discretizations of...

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Scientific computing

Background:

• Advection-diffusion equations are fundamental in modeling various physical phenomena.
• Radial Basis Function Finite Differences (RBF-FD) offer a mesh-free approach for solving differential equations.
• Stabilizing RBF-FD discretizations, especially against spurious oscillations, remains a challenge.

Purpose of the Study:

• To introduce a novel, parameter-free hyperviscosity formulation for stabilizing RBF-FD methods.
• To enhance the robustness and accuracy of RBF-FD solutions for advection-diffusion problems.
• To validate the method's performance on challenging 2D and 3D problems, including a coupled biological model.

Main Methods:

• A quasi-analytical determination of hyperviscosity using 1D semi-discrete Von Neumann analysis.
• Development of a new scaling law for polynomial-augmented RBF-FD methods.
• Integration with a ghost node formulation and the overlapped RBF-FD method.
• Validation using 2D and 3D test cases across a wide range of Peclet numbers (1-1000).

Main Results:

• A generalized, parameter-free hyperviscosity formulation that is efficiently computable.
• Improved robustness and elimination of stagnation errors through a novel ghost node formulation.
• Demonstration of high-order convergence rates for the proposed RBF-FD method in 2D and 3D.
• Successful application to a 3D coupled problem modeling platelet aggregation and coagulation.

Conclusions:

• The novel hyperviscosity formulation effectively stabilizes RBF-FD discretizations of advection-diffusion equations.
• The enhanced RBF-FD method exhibits robust performance and high-order accuracy across various scenarios.
• The developed approach provides a reliable tool for complex simulations in fluid dynamics and biological modeling.