RBF Collocation Methods As Pseudospectral
Methods
Author(s)
G. E. Fasshauer
Abstract
We show how the collocation framework that is prevalent in the radial basis function
literature can be modified so that the methods can be interpreted in the framework
of standard pseudospectral methods. This implies that many of the standard
algorithms and strategies used for solving time-dependent as well as timeindependent
partial differential equations with (polynomial) pseudospectral methods
can be readily adapted for the use with radial basis functions. The potential
advantage of radial basis functions is that they lend themselves to complex geometries
and non-uniform discretizations.
Keywords: radial basis functions, collocation, pseudospectral methods.
1 Pseudospectral methods and radial basis functions
Pseudospectral (PS) methods are known as highly accurate solvers for partial differential
equations (PDEs). The basic idea (see, e.g., [4] or [12]) is to use a set of
(very smooth and global) basis functions φj , j = 1, . . .,N, such as polynomials
to represent an unknown function (the approximate solution of the PDE) via
uh(x) =
N
j=1
λjφj(x), x∈ R. (1)
Since most of our discussion will focus on a representation of the spatial part of
the solution we ignore the time variable in the formulas for uh. We will employ
standard time-stepping procedures to deal with the temporal part of the solution.
Moreover, since standard pseudospectral methods are designed for the univariate
Keywords
radial basis functions, collocation, pseudospectral methods.
Related Book
Other papers in this volume
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