Energy-stable global radial basis function methods on summation-by-parts form

TL;DR

This paper develops energy-stable radial basis function methods for time-dependent PDEs by integrating summation-by-parts operators, addressing stability issues especially with boundary conditions.

Contribution

It introduces a novel framework combining RBF methods with summation-by-parts operators to ensure provable stability for time-dependent PDEs.

Findings

Established energy stability for RBF methods with boundary conditions.

Provided a general framework applicable to various PDEs.

Enhanced reliability of RBF methods in simulations involving time-dependent problems.

Abstract

Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop and construct energy-stable radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite Element communities.