Fourier pseudospectral method on generalized sparse grids for the space-fractional Schrödinger equation

Abstract

In this paper, the time-splitting Fourier pseudospectral method on the generalized sparse grids is applied to solve the space-fractional Schrödinger equation. We give a containment relation between different level-index sets of the generalized sparse grids, and it can be used in designing the reference generalized sparse grids which are finer than other considered grids. Thus the numerical solution on the reference generalized sparse grids can be used as the reference true solution of the equation. Then, the fully discrete algorithm is obtained. In the numerical experiments, we compare the numerical results on the generalized sparse grids with those on the full grids. For the interpolation of the Gaussian multiplied by a factor and for the computation of the Schrödinger equation with two kinds of non-smooth potentials, the advantages of the Fourier pseudospectral method on the generalized sparse grids with the level-index set of parameter K=1,2,3 are manifest in the approximation with high resolution. Here the sparsity of the generalized sparse grids will become weak when the parameter K becomes large. Moreover, the advantage of the generalized sparse grids is more pronounced in solving the Schrödinger equation with the higher dimension, the square well potential or the fractional Laplacian.