Adaptive radial basis function methods for time dependent partial differential equations

Abstract

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.