RBF Collocation Methods As Pseudospectral Methods
Free (open access)
G. E. Fasshauer
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.,  or ) 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
radial basis functions, collocation, pseudospectral methods.