Stability estimates for radial basis function methods applied to linear scalar conservation laws

TL;DR

This work analyzes the time-domain stability of three radial basis function discretizations—Kansa's global RBF, RBF-PUM, and RBF-FD—for linear scalar advection. It derives semi-discrete $\ell_2$-stability estimates, highlighting that oversampling can render Kansa and RBF-PUM stable in time, while RBF-FD requires stabilization of spurious jumps across Voronoi interfaces. A jump-stabilization penalty is proposed to suppress these spurious terms, and numerical experiments confirm the theory: oversampling improves stability for Kansa and RBF-PUM, but RBF-FD stability hinges on both oversampling and jump control. The results provide practical guidance for designing stable, high-order RBF discretizations of hyperbolic PDEs and suggest avenues for extending the framework to nonlinear fluxes and other domains.

Abstract

We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete $\ell_2$-norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be $\ell_2$-stable in time under a sufficiently large oversampling of the discretized system of equations. The RBF-FD method in addition requires stabilization of the spurious jump terms due to the discontinuous RBF-FD cardinal basis functions. Numerical experiments show an agreement with our theoretical observations.

Figures (15)

• Figure 1: Plots from left to right: a point set $X$ (blue markers) obtained using DistMesh, a randomly perturbed DistMesh $X$ point set (blue markers), point set $X$ (blue markers) together with the evaluation point set $Y$ (red markers), where the oversampling parameter is set to $q=4$.
• Figure 2: Nodes (blue points) and different node supports (black points with red edges) specific to three RBF methods: Kansa's RBF method (global support), the RBF partition of unity method (patch support) and the RBF generated finite difference (RBF-FD) method (stencil support). The green area in the RBF-FD method case illustrates the region to which the stencil approximation is further restricted when evaluating the numerical solution.
• Figure 3: Magnitude of the largest jump (discontinuity) in a cardinal basis function $\Psi^*$, as the stencil size is increased. The different curves correspond to the parameter $N$, the number of nodes placed on a 1D domain $\Omega_{\text{1D}}=[0,1]$. For each stencil size and $N$, the observed $\Psi^*$ is chosen such that its center node is closest to the point $x=0.4$.
• Figure 4: One Voronoi cell with a set of midpoints $Y_{\mathcal{E}}$ placed over its edges (red points). The Voronoi cell has one center $X$ point (blue point) and 6 interior evaluation $Y$ points (pale red points).
• Figure 5: Integration error when approximating an integral by means of oversampling, as a function of the inverse internodal distance $1/h_y$ in the evaluation point set $Y$. The three plots are from left to right displaying the convergence when integrating: a Gaussian function, the cubic polyharmonic (PHS) function and a discontinuous function disjointly defined by trigonometric and polynomial functions. The lines in each plot correspond to using three different evaluation point sets $Y$.