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Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

Published online by Cambridge University Press:  03 June 2015

Xu Qian*
Affiliation:
Department of Mathematics and Systems Science, and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, P.R. China Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
Yaming Chen
Affiliation:
Department of Mathematics and Systems Science, and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, P.R. China School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK
Songhe Song
Affiliation:
Department of Mathematics and Systems Science, and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, P.R. China
*
Corresponding author.Email:xq@princeton.edu
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Abstract

In this paper, we propose a wavelet collocation splitting (WCS) method, and a Fourier pseudospectral splitting (FPSS) method as comparison, for solving one-dimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics. The two methods can preserve the intrinsic properties of original problems as much as possible. The splitting technique increases the computational efficiency. Meanwhile, the error estimation and some conservative properties are investigated. It is proved to preserve the charge conservation exactly. The global energy and momentum conservation laws can be preserved under several conditions. Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2014

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