Search

We approximate the solution for the time dependent Schrödinger equation in two steps. We first use a pseudospectral collocation method that uses samples of the functions on rank-1 or rank-r lattice points. We then get a system of ordinary differential equations in time, which we solve approximately by stepping in time using the Strang splitting method. We prove that the numerical scheme proposed converges quadratically with respect to the time step size, given that the potential is in a Korobov space with the smoothness parameter greater than 9/2. Particularly, we prove that the required degree of smoothness is independent of the dimension of the problem. We demonstrate our new method by comparing with results using sparse grids from [V. Gradinaru, SIAM J. Numer. Anal., 46 (2007), pp. 103--123], with several numerical examples showing the large advantage for our new method and pushing the examples to higher dimensionality. The proposed method has two distinctive features from a numerical perspective: (i) numerical results show the error convergence of time discretization is consistent even for higher-dimensional problems; (ii) by using the rank-1 lattice points, the solution can be efficiently computed (and further time stepped) using only one-dimensional fast Fourier transforms. [ABSTRACT FROM AUTHOR]