Exploring Oversampling in RBF Least-Squares Collocation Method of Lines for Surface Diffusion

TL;DR

This paper analyzes how oversampling affects the stability and accuracy of the RBF-LSC method of lines for surface diffusion, providing guidelines for optimal oversampling ratios to improve numerical performance.

Contribution

It offers a detailed numerical investigation into the role of oversampling in RBF-LSC-MoL, highlighting its impact on stability and accuracy for surface diffusion problems.

Findings

Oversampling ratio significantly influences eigenvalue stability.

Optimal oversampling balances accuracy and computational efficiency.

Guidelines for selecting oversampling ratios are provided.

Abstract

This paper investigates the numerical behavior of the radial basis functions least-squares collocation (RBF-LSC) method of lines (MoL) for solving surface diffusion problems, building upon the theoretical analysis presented in [SIAM J. Numer. Anal., 61 (3), 1386-1404}]. Specifically, we examine the impact of the oversampling ratio, defined as the number of collocation points used over the number of RBF centers for quasi-uniform sets, on the stability of the eigenvalues, time stepping sizes taken by Runge-Kutta methods, and overall accuracy of the method. By providing numerical evidence and insights, we demonstrate the importance of the oversampling ratio for achieving accurate and efficient solutions with the RBF-LSC-MoL method. Our results reveal that the oversampling ratio plays a critical role in determining the stability of the eigenvalues, and we provide guidelines for selecting an optimal oversampling ratio that balances accuracy and computational efficiency.