Spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries. Since radial basis function (RBF) approximations permit this even for multivariate scattered data, there has been much recent interest in practical algorithms to compute these approximations effectively.Several types of RBFs feature a free parameter (e.g., c in the multiquadric (MQ) case ¢(r) = ~).The limit of c ~ ~ (increasingly flat basis functions) has not received much attention because it leads to a severely ill-conditioned problem. We present here an algorithm which avoids this difficulty, and which allows numerically stable computations of MQ RBF interpolants for all parameter values. We then find that the accuracy of the resulting approximations, in some cases, becomes orders of magnitude higher than was the case within the previously available parameter range.Our new method provides the first tool for the numerical exploration of MQ RBF interpolants in the limit of c --* c~. The method is in no way specific to MQ basis functions and can--without any change---be applied to many other cases as well. (~)

. 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.

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.

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.

In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on R d , such as radial basis functions (RBFs), to a smooth, compact embedded submanifold M ⊂ R d . For restricted kernels having finite smoothness, we provide a complete characterization of the native space on M. After this and some preliminary setup, we present Sobolev-type error estimates for the interpolation problem. Numerical results verifying the theory are also presented for a one-dimensional curve embedded in R 3 and a two-dimensional torus.

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In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in R d . For two-dimensional surfaces embedded in R 3 , these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented.

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