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

Wright, Grady B.; Fornberg, Bengt

doi:10.1016/j.jcp.2005.05.030

Cited by 278 publications

(137 citation statements)

References 36 publications

(64 reference statements)

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“…However, it is only in the last 15 years that it has been applied to solving mixed partial differential equations (PDEs) containing parabolic and/or elliptic operators (cf. [1][2][3][4][5][6]). It has furthermore only been considered for PDEs in spherical domains for these same operators in the last 5 years [7,8].…”

Section: Introductionmentioning

confidence: 99%

Transport schemes on a sphere using radial basis functions

Flyer

Wright

2007

Journal of Computational Physics

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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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Section: Introductionmentioning

confidence: 99%

Transport schemes on a sphere using radial basis functions

Flyer

Wright

2007

Journal of Computational Physics

Self Cite

133

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“…For any scattered nodes, the restriction of the structured grid is overcome by setting the approximation of the function derivative to be a linear combination of function values on scattered nodes in the stencil; however, the methodology for computing the coefficients of finite difference formulas for any scattered points with dimensions of more than one has the problem of well-posedness for polynomial interpolation [16]. Thus, the combination of RBF and finite difference methodology (RBF-FD) is introduced to overcome the well-posedness problem.…”

Section: Rbf-fd Methodsmentioning

confidence: 99%

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

Yensiri

Skulkhu

2017

Mathematics

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Abstract:The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies.

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“…We note that the RBF-FD method has proven successful for a number of other applications in planar domains in two and higher dimensions (e.g. [27][28][29][30][31][32]), and on the surface of a sphere [33,34], but that this is the first application of the method to more general 1D surfaces (curves).…”

Section: Approximating the Surface Laplacianmentioning

confidence: 99%

“…It has also been shown through experience and studies [30,33] that better accuracy is gained by additionally requiring that the linear combination (13) be exact for a constant. Hence, we also impose the following constraint on the weights γ i :…”

Section: Approximating the Surface Laplacianmentioning

confidence: 99%