Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs
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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 ℓ_2-norm intrinsic to each of the three methods.The results show that Kansa's method and RBF-PUM can be ℓ_2-stable in time under a sufficiently large oversampling of the discretized system of equations. On the other hand, the RBF-FD method is not ℓ_2-stable by construction, no matter how large the oversampling is. We show that this is due to the jumps (discontinuities) in the RBF-FD cardinal basis functions. We also provide a stabilization of the RBF-FD method that penalizes the spurious jumps. Numerical experiments show an agreement with our theoretical observations.
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