Communications in Information and Systems

Volume 13 (2013)

Number 3

Special Issue in Honor of Marshall Slemrod: Part 3 of 4

A semi-Lagrangian time splitting method for the Schrödinger equation with vector potentials

Pages: 247 – 289

DOI: http://dx.doi.org/10.4310/CIS.2013.v13.n3.a1

Authors

Shi Jin (Dept. of Mathematics, Institute of Nature Science and Ministry of Education Key Laboratory, Scientific & Engineering Computing, Shanghai Jiao Tong University, Shanghai, China; Dept. of Mathematics, Univ. of Wisconsin, Madison, Wisc., U.S.A.)

Zhennan Zhou (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Abstract

In this paper, we present a time splitting scheme for the Schrödinger equation in the presence of electromagnetic field in the semi-classical regime, where the wave function propagates $O(\epsilon)$ oscillations in space and time. With the operator splitting technique, the time evolution of the Schrödinger equation is divided into three parts: the kinetic step, the convection step and the potential step. The kinetic and the potential steps can be handled by the classical timesplitting spectral method. For the convection step, we propose a semi-Lagrangian method in order to allow large time steps. We prove the unconditional stability conditions with spatially variant external vector potentials, and the error estimate in the $l^2$ approximation of the wave function. By comparing with the semi-classical limit, the classical Liouville equation in the Wigner framework, we show that this method is able to capture the correct physical observables with time step $\Delta t \gg \epsilon$. We implement this method numerically for both one dimensional and two dimensional cases to verify that $\epsilon$-independent time steps can indeed be taken in computing physical observables.

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