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

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 discr...

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Veröffentlicht in:Applied mathematics and computation Jg. 485; S. 129020
Hauptverfasser: Tominec, Igor, Nazarov, Murtazo, Larsson, Elisabeth
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 15.01.2025
Schlagworte:
Beräkningsvetenskap med inriktning mot numerisk analys
Hyperbolic PDE
Kansa
Radial basis function
RBF-FD
RBF-PUM
Scientific Computing with specialization in Numerical Analysis
Stability
Radial basis function
Kansa
RBF-FD
Stability
Hyperbolic PDE
RBF-PUM
ISSN:0096-3003, 1873-5649
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Zusammenfassung: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.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2024.129020