Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions

TL;DR

This paper addresses instability in radial basis function methods for hyperbolic conservation laws caused by boundary condition enforcement, proposing a weak enforcement approach that improves stability and conservation.

Contribution

The paper introduces a weak enforcement technique for boundary conditions in RBF methods, enhancing stability and conservation in solving hyperbolic conservation laws.

Findings

Weak enforcement of BCs stabilizes RBF methods.

The approach preserves conservation properties.

Numerical experiments confirm theoretical improvements.

Abstract

It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs). Here we investigate this phenomenon and identify the strong enforcement of BCs as the mechanism triggering such stability issues. Based on this observation we propose a technique to weakly enforce BCs in RBF methods. In the case of hyperbolic CLs, this is achieved by carefully building RBF methods from the weak form of the CL, rather than the typically enforced strong form. Furthermore, we demonstrate that global RBF methods may violate conservation, yielding physically unreasonable solutions when the approximation does not take into account these considerations. Numerical experiments validate our theoretical results.