Highly Localized RBF Lagrange Functions for Finite Difference Methods on Spheres

TL;DR

This paper demonstrates how rapidly decaying RBF Lagrange functions on spheres enable the development of stable and accurate finite difference methods for PDEs, with proven convergence properties.

Contribution

It introduces a novel approach using localized RBF Lagrange functions for finite difference methods on spheres, providing stability and convergence guarantees.

Findings

Effective finite difference schemes for PDEs on spheres.

Stable and accurate discretizations with moderate stencil growth.

Convergence estimates for specific PDE classes.

Abstract

The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the spheres can be used to create effective, stable finite difference methods based on radial basis functions (RBF-FD). For certain classes of PDEs this approach leads to precise convergence estimates for stencils which grow moderately with increasing discretization fineness.