A linear system-free Gaussian RBF method for the Gross-Pitaevskii equation on unbounded domains



Gaussian radial basis function (RBF) interpolation methods are theoretically spectrally accurate. However, in applications this accuracy is seldom realized due to the necessity of solving a very poorly conditioned linear system to evaluate the methods. Recently, by using approximate cardinal functions and restricting the method to a uniformly spaced grid (or a smooth mapping thereof), it has been shown that the Gaussian RBF method can be formulated in a matrix free framework that does not involve solving a linear system [1]. In this work, we differentiate the linear system-free Gaussian (LSFG) method and use it to solve partial differential equations on unbounded domains that have solutions that decay rapidly and that are negligible at the ends of the grid. As an application, we use the LSFG collocation method to numerically simulate Bose-Einstein condensates. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 389–401, 2012