Stability analysis of RBF-FD and WLS based local strong form meshless methods on scattered nodes

TL;DR

This paper compares the stability and accuracy of RBF-FD and WLS meshless methods on scattered nodes, highlighting their respective advantages for different approximation orders in 2D and 3D problems.

Contribution

It provides a detailed stability and accuracy analysis of RBF-FD and WLS methods, revealing their suitability for various approximation orders and computational costs.

Findings

WLS performs better for lower order approximations.

RBF-FD offers higher accuracy and stability for higher order approximations.

RBF-FD has higher computational complexity.

Abstract

The popularity of local meshless methods in the field of numerical simulations has increased greatly in recent years. This is mainly due to the fact that they can operate on scattered nodes and that they allow a direct control over the approximation order and basis functions. In this paper we analyse two popular variants of local strong form meshless methods, namely the radial basis function-generated finite differences (RBF-FD) using polyharmonic splines (PHS) augmented with monomials, and the weighted least squares (WLS) approach using only monomials. Our analysis focuses on the accuracy and stability of the numerical solution computed on scattered nodes in a two- and three-dimensional domain. We show that while the WLS variant is a better choice when lower order approximations are sufficient, the RBF-FD variant exhibits a more stable behavior and a higher accuracy of the numerical solution for higher order approximations, but at the cost of higher computational complexity.