Transport schemes on a sphere using radial basis functions

Flyer, Natasha; Wright, Grady B.

doi:10.1016/j.jcp.2007.05.009

Cited by 97 publications

(138 citation statements)

References 37 publications

(63 reference statements)

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“…It has previously been shown that global radial basis functions (RBF) are highly competitive with respect to other state-of-the-art numerical methods in the arena of computational geoscience [15][16][17][18]46]. However, they become computationally expensive when scaled to very large numbers of nodes.…”

Section: Introductionmentioning

confidence: 99%

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

Flyer

Lehto

Blaise

et al. 2012

Journal of Computational Physics

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158

173

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. AbstractThe current paper establishes the computational efficiency and accuracy of the RBF-FD method for large-scale geoscience modeling with comparisons to state-of-the-art methods as high-order discontinuous Galerkin and spherical harmonics, the latter using expansions with close to 300,000 bases. The test cases are demanding fluid flow problems on the sphere that exhibit numerical challenges, such as Gibbs phenomena, sharp gradients, and complex vortical dynamics with rapid energy transfer from large to small scales over short time periods. The computations were possible as well as very competitive due to the implementation of hyperviscosity on large RBF stencil sizes (corresponding roughly to 6th to 9th order methods) with up to O(10 5 ) nodes on the sphere. The RBF-FD method scaled as O(N ) per time step, where N is the total number of nodes on the sphere. In the Appendix, guidelines are given on how to chose parameters when using RBF-FD to solve hyperbolic PDEs.

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Section: Introductionmentioning

confidence: 99%

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

Flyer

Lehto

Blaise

et al. 2012

Journal of Computational Physics

Self Cite

158

173

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“…A future goal therefore is to combine the treecode with a method to overcome the ill-conditioning in that limit [3,12,15,17,18]. If this can be accomplished, the resulting RBF method may lead to advances in various applications, including geophysical fluid flow [16].…”

Section: Discussionmentioning

confidence: 99%

“…In the first case we consider random nodes in a unit cube, and in the second case the nodes were projected radially from the cube to the surface of a unit sphere. One motivation for considering the sphere is the growing interest in RBF methods for geophysical fluid flow [16]. In both test cases, the RBF coefficients were set to λ i = 1, as in [10], and the RBF parameter was set to c = 10 −1 .…”

Section: Description Of Treecodementioning

confidence: 99%

Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode

Krasny¹,

Wang²

2011

SIAM J. Sci. Comput.

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Abstract.A treecode is presented for evaluating sums defined in terms of the multiquadric radial basis function (RBF), φ(x) = (|x| 2 + c 2 ) 1/2 , where x ∈ R 3 and c ≥ 0. Given a set of N nodes, evaluating an RBF sum directly requires CPU time that scales like O(N 2 ). For a given level of accuracy, the treecode reduces the CPU time to O(N log N ) using a far-field expansion of φ(x). We consider two options for the far-field expansion: (1) a Laurent series previously used in applications of the Fast Multipole Method to multiquadric RBFs, and (2) a certain Taylor series previously used in treecode particle simulations, but not yet in the context of multiquadric RBFs. It is known that the Laurent series converges when the RBF parameter c lies in an interval 0 ≤ c ≤c, wherec is proportional to the minimum node spacing, but here we show that the Taylor series converges uniformly for c ≥ 0. We implement the treecode in Cartesian coordinates and use a recurrence relation to compute the Taylor coefficients. Numerical results exhibit the treecode error, CPU time, and memory usage in two test cases, random nodes in a cube and on the surface of a sphere. The treecode approach presented here is applicable to generalized multiquadrics in any dimension.

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“…For other functions, the relation is an approximation. Differentiation matrices are needed when solving PDEs using global RBF approximations [24,8,5,23,6] or solving PDEs using partition of unity based RBF methods [20]. Stencil weights are a special case of differentiation matrix, where the derivative value only at one (often central) node point of the stencil is considered.…”

Section: Introduction Radial Basis Function (Rbf) Approximationmentioning

confidence: 99%

Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions

Larsson¹,

Lehto²,

Heryudono³

et al. 2013

SIAM J. Sci. Comput.

119

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Abstract. Radial basis function (RBF) approximation has the potential to provide spectrally accurate function approximations for data given at scattered node locations. For smooth solutions, the best accuracy for a given number of node points is typically achieved when the basis functions are scaled to be nearly flat. This also results in nearly linearly dependent basis functions and severe ill-conditioning of the interpolation matrices. Fornberg, Larsson, and Flyer recently generalized the RBF-QR method to provide a numerically stable approach to interpolation with flat and nearly flat Gaussian RBFs for arbitrary node sets in up to three dimensions. In this work, we consider how to extend this method to the task of computing differentiation matrices and stencil weights in order to solve partial differential equations. The expressions for first and second order derivative operators as well as hyperviscosity operators are established, numerical issues such as how to deal with non-unisolvency are resolved, and the accuracy and computational efficiency of the method are tested numerically. The results indicate that using the RBF-QR approach for solving PDE problems can be very competitive compared with using the ill-conditioned direct solution approach or using variable precision arithmetic to overcome the conditioning issue.Key words. radial basis function, flat limit, ill-conditioning, differentiation matrix, stencil weight, RBF, RBF-QR, RBF-FD AMS subject classifications. 65D15, 65D251. Introduction. Radial basis function (RBF) approximation [1,31,4] is emerging as an important method class for interpolation, approximation, and solution of partial differential equations (PDEs) for data given at scattered node locations, with non-trivial geometry, or with computational domains in higher dimensions. The main advantages are the spectral convergence rates that can be achieved using infinitely smooth basis functions, the geometrical flexibility, and the ease of implementation. However, in practical cases, convergence has often been hampered by ill-conditioning as the shape of the basis functions become flatter. The best accuracy for smooth and well resolved solutions is often found in this regime [18,13,19]. Therefore, moving to larger shape parameter values (less flat RBFs) is not a desirable solution to the conditioning problem.The first method that allowed stable computations in the flat RBF regime was the Contour-Padé method derived by Fornberg and Wright [13]. The method works in any number of dimensions, but for relatively low numbers of nodes. Except for the approach in [24], limited to Gaussian RBFs on an equispaced grid in one dimension, the next method that was developed was the RBF-QR method, which was first derived for nodes on the surface of the sphere by Fornberg and Piret [12] and then for general node distributions in up to three dimensions [9]. The RBF-QR methods can be employed for approximations over thousands of nodes.In [9], we gave examples of convergence and performance results in the case of i...

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Transport schemes on a sphere using radial basis functions

Cited by 97 publications

References 37 publications

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode

Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions

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