Grady B. Wright

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

In standard equispaced finite difference (FD) formulas, symmetries can make the order of accuracy relatively high compared to the number of nodes in the FD stencil. With scattered nodes, such symmetries are no longer available. The generalization of compact FD formulas that we propose for scattered nodes and radial basis functions (RBFs) achieves the goal of still keeping the number of stencil nodes small without a similar reduction in accuracy. We analyze the accuracy of these new compact RBF-FD formulas by applying them to some model problems, and study the effects of the shape parameter that arises in, for example, the multiquadric radial function.

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

The 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 Appendix A, guidelines are given on how to chose parameters when using RBF-FD to solve hyperbolic PDEs.

Transport Schemes on a Sphere Using Radial Basis Functions

The 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.

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

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.

A new class of oscillatory radial basis functions

Radial basis functions (RBFs) form a primary tool for multivariate interpolation, and they are also receiving increased attention for solving PDEs on irregular domains. Traditionally, only nonoscillatory radial functions have been considered. We find here that a certain class of oscillatory radial functions (including Gaussians as a special case) leads to nonsingular interpolants with intriguing features especially as they are scaled to become increasingly flat. This flat limit is important in that it generalizes traditional spectral methods to completely general node layouts. Interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions (such as multiquadrics and inverse multiquadratics).

A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction---Diffusion Equations on Surfaces

In this paper, we present a method based on radial basis function (RBF)-generated finite differences (FD) for numerically solving diffusion and reaction–diffusion equations (PDEs) on closed surfaces embedded in

A hybrid radial basis function–pseudospectral method for thermal convection in a 3‐D spherical shell

{\mathbb {R}}^d

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

Rd. Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.