A simple adaptive technique for nonlinear wave problems
Journal of Computational Physics
SIAM Journal on Numerical Analysis
An adaptive pseudo-spectral method for reaction diffusion problems
Journal of Computational Physics
An adaptive pseudospectral method for discontinuous problems
Applied Numerical Mathematics
Journal of Computational Physics
Family of spectral filters for discontinuous problems
Journal of Scientific Computing
A modified Chebyshev pseudospectral method with an O(N–1) time step restriction
Journal of Computational Physics
A moving collocation method for solving time dependent partial differential equations
Applied Numerical Mathematics
Error estimates for interpolation by compactly supported radial basis functions of minimal degree
Journal of Approximation Theory
A Wavelet-Optimized, Very High Order Adaptive Grid and Order Numerical Method
SIAM Journal on Scientific Computing
On the numerical solution of one-dimensional PDEs using adaptive methods based on equidistribution
Journal of Computational Physics
Radial Basis Functions
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
Digital Total Variation Filtering as Postprocessing for Radial Basis Function Approximation Methods
Computers & Mathematics with Applications
Computers & Mathematics with Applications
Radial function collocation solution of partial differential equations in irregular domains
International Journal of Computing Science and Mathematics
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Radial basis function (RBF) methods have shown the potential to be a universal grid free method for the numerical solution of partial differential equations. Both global and compactly supported basis functions may be used in the methods to achieve a higher order of accuracy. In this paper, we take advantage of the grid free property of the methods and use an adaptive algorithm to choose the location of the collocation points. The RBF methods produce results similar to the more well-known and analyzed spectral methods, but while allowing greater flexibility in the choice of grid point locations. The adaptive RBF methods are most successful when the basis functions are chosen so that the PDE solution can be approximated well with a small number of the basis functions.