Stability estimates for radial basis function methods applied to linear scalar conservation laws - Tominec, Igor; Nazarov, Murtazo; Larsson, Elisabeth (Applied mathematics and computation, 2025)
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 ℓ2-norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be ℓ2-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. •Stability estimates for RBF methods discretizing linear conservation laws.•Spurious terms in Kansa's method, RBF-PUM, RBF-FD arise from lack of exact quadrature.•The RBF-FD estimate also entails a spurious term due to discontinuous trial space.•Oversampling and jump stabilization are possible stabilizations.