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

Authors:
Grady B. Wright;Bengt Fornberg
Affiliations:
Department of Mathematics, University of Utah, 233 Widstoe Building, Salt Lake City, UT 84112-0090, USA;Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA
Venue:
Journal of Computational Physics
Year:
2006

Citing 10
Cited 20

On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation

Journal of Computational Physics
Applied numerical linear algebra

Applied numerical linear algebra
An efficient numerical scheme for Burgers' equation

Applied Mathematics and Computation
Classroom Note: Calculation of Weights in Finite Difference Formulas

SIAM Review
Iterative solution of nonlinear equations in several variables

Iterative solution of nonlinear equations in several variables
Radial basis function interpolation: numerical and analytical developments

Radial basis function interpolation: numerical and analytical developments
Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions

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
Numerical comparison of least square-based finite-difference (LSFD) and radial basis function-based finite-difference (RBFFD) methods

Computers & Mathematics with Applications

Compact finite difference method for American option pricing

Journal of Computational and Applied Mathematics
Compact finite volume schemes on boundary-fitted grids

Journal of Computational Physics
Local radial basis function based gridfree scheme for unsteady incompressible viscous flows

Journal of Computational Physics
The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems

Journal of Computational Physics
A singularity-avoiding moving least squares scheme for two-dimensional unstructured meshes

Journal of Computational Physics
Voxel Based Adaptive Meshless Method for Cardiac Electrophysiology Simulation

FIMH '09 Proceedings of the 5th International Conference on Functional Imaging and Modeling of the Heart
Discretization correction of general integral PSE Operators for particle methods

Journal of Computational Physics
RBF-FD formulas and convergence properties

Journal of Computational Physics
Adaptive meshless centres and RBF stencils for Poisson equation

Journal of Computational Physics
Stabilization of RBF-generated finite difference methods for convective PDEs

Journal of Computational Physics
Optimal constant shape parameter for multiquadric based RBF-FD method

Journal of Computational Physics
On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

Computers & Mathematics with Applications
Stable Computations with Gaussian Radial Basis Functions

SIAM Journal on Scientific Computing
Optimal variable shape parameter for multiquadric based RBF-FD method

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

Journal of Computational Physics
Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs

Journal of Computational Physics
A compact five-point stencil based on integrated RBFs for 2D second-order differential problems

Journal of Computational Physics
Stable calculation of Gaussian-based RBF-FD stencils

Computers & Mathematics with Applications
Application of radial basis function with L-stable Padé time marching scheme for pricing exotic option

Computers & Mathematics with Applications
An alternative local collocation strategy for high-convergence meshless PDE solutions, using radial basis functions

Journal of Computational Physics

Quantified Score

Hi-index	31.53

Visualization

Abstract

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.