Spectral Approximation Orders of Radial Basis Function Interpolation on the Sobolev Space

Yoon, Jungho

doi:10.1137/s0036141000373811

Cited by 101 publications

(49 citation statements)

References 16 publications

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“…and ð1C ð3rÞ 2 Þ K1 , will lead to spectral convergence (Madych & Nelson 1992;Yoon 2001), as is demonstrated in this paper. Note that the infinitely smooth RBFs depend on a shape parameter 3.…”

Section: Introduction To Rbfssupporting

confidence: 63%

A radial basis function method for the shallow water equations on a sphere

2009

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The paper derives the first known numerical shallow water model on the sphere using radial basis function (RBF) spatial discretization, a novel numerical methodology that does not require any grid or mesh. In order to perform a study with regard to its spatial and temporal errors, two nonlinear test cases with known analytical solutions are considered. The first is a global steady-state flow with a compactly supported velocity field, while the second is an unsteady flow where features in the flow must be kept intact without dispersion. This behaviour is achieved by introducing forcing terms in the shallow water equations. Error and time stability studies are performed, both as the number of nodes are uniformly increased and the shape parameter of the RBF is varied, especially in the flat basis function limit. Results show that the RBF method is spectral, giving exceptionally high accuracy for low number of basis functions while being able to take unusually large time steps. In order to put it in the context of other commonly used global spectral methods on a sphere, comparisons are given with respect to spherical harmonics, double Fourier series and spectral element methods.

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Section: Introduction To Rbfssupporting

confidence: 63%

A radial basis function method for the shallow water equations on a sphere

2009

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“…In this study, we are primarily interested in the infinitely smooth radial functions since, by suitable choices of the shape parameter ε, they can provide more accurate interpolants than the piecewise smooth case [26,27]. We postpone the discussion on the effect of ε until Section 4.…”

Section: Standard Rbf Interpolationmentioning

confidence: 99%

Scattered node compact finite difference-type formulas generated from radial basis functions

Wright,

Fornberg

2006

Journal of Computational Physics

276

130

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“…[13]). On the other hand, the evidence strongly suggests that infinitely smooth RBFs will lead to spectral convergence [14,15]. Notice that the infinitely smooth RBFs depend on a shape parameter ε.…”

Section: Introduction To Radial Basis Functionsmentioning

confidence: 99%

Transport schemes on a sphere using radial basis functions

Flyer

Wright

2007

Journal of Computational Physics

133

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This is an author-produced, peer-reviewed version of this article. AbstractThe aim of this work is to introduce the physics community to the high performance of radial basis functions (RBFs) compared to other spectral methods for modeling transport (pure advection) and to provide the first known application of the RBF methodology to hyperbolic partial differential equations on a sphere. First, it is shown that even when the advective operator is posed in spherical coordinates (thus having singularities at the poles), the RBF formulation of it is completely singularity-free. Then, two classical test cases are conducted: 1) linear advection, where the initial condition is simply transported around the sphere and 2) deformational flow (idealized cyclogenesis), where an angular velocity is applied to the initial condition, spinning it up around an axis of rotation. The results show that RBFs allow for a much lower spatial resolution (i.e. lower number of nodes) while being able to take unusually large time-steps to achieve the same accuracy as compared to other commonly used spectral methods on a sphere such as spherical harmonics, double Fourier series, and spectral element methods. Furthermore, RBFs are algorithmically much simpler to program.

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Spectral Approximation Orders of Radial Basis Function Interpolation on the Sobolev Space

Cited by 101 publications

References 16 publications

A radial basis function method for the shallow water equations on a sphere

A radial basis function method for the shallow water equations on a sphere

Scattered node compact finite difference-type formulas generated from radial basis functions

Transport schemes on a sphere using radial basis functions

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