Strang splitting in combination with rank-$1$ and rank-$r$ lattices for the time-dependent Schr\"odinger equation

TL;DR

This paper introduces a new numerical method combining Strang splitting with rank-1 and rank-r lattice sampling for solving the time-dependent Schrödinger equation, achieving high accuracy and efficiency in high dimensions.

Contribution

The paper develops a novel approach that integrates lattice-based spectral collocation with Strang splitting, providing dimension-independent smoothness requirements and efficient computation for high-dimensional TDSE.

Findings

Quadratic convergence of the numerical scheme with respect to time step size.

Efficient computation using 1D FFTs due to lattice sampling.

Superior performance and higher dimensional applicability compared to sparse grid methods.

Abstract

We approximate the solution for the time dependent Schr\"odinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of functions on rank-1 or rank-r lattice points with unitary Fourier transforms. 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 [12], with several numerical examples showing 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 $1$-dimensional Fast Fourier Transforms.