Scattered node compact finite difference-type formulas generated from radial basis functions
Journal of Computational Physics
The Runge phenomenon and spatially variable shape parameters in RBF interpolation
Computers & Mathematics with Applications
On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere
Journal of Computational Physics
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems
Journal of Computational Physics
Computers & Mathematics with Applications
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
A new class of oscillatory radial basis functions
Computers & Mathematics with Applications
Stabilization of RBF-generated finite difference methods for convective PDEs
Journal of Computational Physics
Computers & Mathematics with Applications
Stable Computations with Gaussian Radial Basis Functions
SIAM Journal on Scientific Computing
Journal of Computational Physics
Stable Evaluation of Gaussian Radial Basis Function Interpolants
SIAM Journal on Scientific Computing
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Traditional finite difference (FD) methods are designed to be exact for low degree polynomials. They can be highly effective on Cartesian-type grids, but may fail for unstructured node layouts. Radial basis function-generated finite difference (RBF-FD) methods overcome this problem and, as a result, provide a much improved geometric flexibility. The calculation of RBF-FD weights involves a shape parameter @e. Small values of @e (corresponding to near-flat RBFs) often lead to particularly accurate RBF-FD formulas. However, the most straightforward way to calculate the weights (RBF-Direct) becomes then numerically highly ill-conditioned. In contrast, the present algorithm remains numerically stable all the way into the @e-0 limit. Like the RBF-QR algorithm, it uses the idea of finding a numerically well-conditioned basis function set in the same function space as is spanned by the ill-conditioned near-flat original Gaussian RBFs. By exploiting some properties of the incomplete gamma function, it transpires that the change of basis can be achieved without dealing with any infinite expansions. Its strengths and weaknesses compared with the Contour-Pade, RBF-RA, and RBF-QR algorithms are discussed.