Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation
Journal of Approximation Theory
Spectral transform solutions to the shallow water test set
Journal of Computational Physics
The spectral element method for the shallow water equations on the sphere
Journal of Computational Physics
Fast Shallow-Water equation solvers in latitude-longitude coordinates
Journal of Computational Physics
Spectral methods in MATLAB
Radial Basis Functions
The NCAR Spectral Element Climate Dynamical Core: Semi-Implicit Eulerian Formulation
Journal of Scientific Computing
The Runge phenomenon and spatially variable shape parameters in RBF interpolation
Computers & Mathematics with Applications
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
Computers & Mathematics with Applications
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
A new class of oscillatory radial basis functions
Computers & Mathematics with Applications
Eigenvalue stability of radial basis function discretizations for time-dependent problems
Computers & Mathematics with Applications
Journal of Computational Physics
Three ways to solve the Poisson equation on a sphere with Gaussian forcing
Journal of Computational Physics
Application of the RBF meshless method to the solution of the radiative transport equation
Journal of Computational Physics
An alternative procedure for selecting a good value for the parameter c in RBF-interpolation
Advances in Computational Mathematics
Stabilization of RBF-generated finite difference methods for convective PDEs
Journal of Computational Physics
Stable Computations with Gaussian Radial Basis Functions
SIAM Journal on Scientific Computing
CAD and mesh repair with Radial Basis Functions
Journal of Computational Physics
International Journal of Computer Applications in Technology
Journal of Computational Physics
Journal of Computational Physics
Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs
Journal of Computational Physics
Stable calculation of Gaussian-based RBF-FD stencils
Computers & Mathematics with Applications
A radial basis functions method for fractional diffusion equations
Journal of Computational Physics
Advances in Computational Mathematics
Hi-index | 31.49 |
Radial basis function (RBF) approximations have been used for some time to interpolate data on a sphere (as well as on many other types of domains). Their ability to solve, to spectral accuracy, convection-type PDEs over a sphere has been demonstrated only very recently. In such applications, there are two main choices that have to be made: (i) which type of radial function to use, and (ii) what value to choose for their shape parameter (denoted by @e, and with flat basis functions - stretched out in the radial direction - corresponding to @e=0). The recent RBF-QR algorithm has made it practical to compute stably also for small values of @e. Results from solving a convective-type PDE on a sphere are compared here for many choices of radial functions over the complete range of @e-values (from very large down to the limit of @e-0). The results are analyzed with a methodology that has similarities to the customary Fourier analysis in equispaced 1-D periodic settings. In particular, we find that high accuracy can be maintained also over very long time integrations. We furthermore gain insights into why RBFs sometimes offer higher accuracy than spherical harmonics (since the latter arise as an often non-optimal special case of the former). Anticipated future application areas for RBF-based methods in spherical geometries include weather and climate modeling.