Commun. Comput. Phys., 15 (2014), pp. 692-711.


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

Xu Qian 1*, Yaming Chen 2, Songhe Song 3

1 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.
2 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.
3 Department of Mathematics and Systems Science, and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, P.R. China.

Received 12 March 2013; Accepted (in revised version) 2 August 2013
Available online 1 November 2013
doi:10.4208/cicp.120313.020813a

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 Schrodinger 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.

AMS subject classifications: 37M05, 65M70, 65T60

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Key words: Schrodinger equation, wavelet collocation method, splitting technique, conservative property.

*Corresponding author.
Email: xq@princeton.edu (X. Qian), yaming.chen@qmul.ac.uk (Y. M. Chen), shsong@nudt.edu.cn (S. H. Song)
 

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