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

Abstract

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.

Augmenting the immersed boundary method with Radial Basis Functions (RBFs) for the modeling of platelets in hemodynamic flows

“…Several studies have provided error estimates for RBF interpolation on circles and spheres; in fact, these interpolants can provide spectral accuracy provided the underlying target function is sufficiently smooth [27,28]. The RBF method has also been used successfully for numerically solving partial differential equations on the surface of the sphere [29,30], as well as more general surfaces [20,31].…”

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

“…Two other potential remedies are increasing ε for the nodes where no suitable stencils can be found, and refining the node set locally in these areas. The former has previously been noted to improve stability (see, e.g., [16]), but larger values of ε also tend to reduce the accuracy. A full exploration of these latter two approaches is beyond the scope of this work.…”

“…Global radial basis function (RBF) methods are quite popular for the numerical solution of various partial differential equations (PDEs) due to their ability to handle scattered node layouts, their simplicity of implementation, and their potential for spectral accuracy for smooth problems. These methods have been successfully applied to the solution of PDEs in various geometries in R 2 and R 3 (e.g., [12,17]), including spherical domains (e.g., [27,16,42]), and more general surfaces embedded in R 3 (e.g., [26,36]). When high orders of algebraic accuracy are sufficient for a given problem, or if the solutions to the problem are expected to only have finite smoothness, RBF generated finite difference (RBF-FD) formulas are an attractive alternative to global RBFs as they perform better in terms of accuracy per computational cost [17].…”

Efficient solvers for the shallow water equations on a sphere

“…Let us now introduce the space of rbfs S In our numerical simulations we use multiquadric radial basis functions with ε = 3.25 , as those used by Flyer [5].…”

“…When the given data (that is, initial conditions) involve scattered data, radial basis functions (rbfs) are especially suitable to approximate the solutions of the pdes as they do not require any mesh generation. Recently a collocation method using rbfs was proposed for the spherical swes [5]. Another possible way to deal with scattered data is to use spherical splines [1,3] (in the sense of Schumaker).…”