Towards Stable Radial Basis Function Methods for Linear Advection Problems

TL;DR

This paper develops two new radial basis function methods with weak boundary enforcement for linear advection problems, ensuring energy stability and demonstrating effectiveness through numerical experiments.

Contribution

It introduces two novel RBF approaches based on flux reconstruction and SATs that achieve strong energy stability for linear advection problems.

Findings

Both methods are proven to be strongly energy stable.

Numerical results confirm the theoretical stability in 1D and 2D cases.

The approaches effectively handle boundary conditions without stability issues.

Abstract

In this work, we investigate (energy) stability of global radial basis function (RBF) methods for linear advection problems. Classically, boundary conditions (BC) are enforced strongly in RBF methods. By now it is well-known that this can lead to stability problems, however. Here, we follow a different path and propose two novel RBF approaches which are based on a weak enforcement of BCs. By using the concept of flux reconstruction and simultaneous approximation terms (SATs), respectively, we are able to prove that both new RBF schemes are strongly (energy) stable. Numerical results in one and two spatial dimensions for both scalar equations and systems are presented, supporting our theoretical analysis.