LOD-MS for Gross-Pitaevskii Equation in Bose-Einstein Condensates
Linghua Kong 1*, Jialin Hong 2, Jingjing Zhang 31 School of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, P.R. China.
2 State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, P.R. China.
3 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, P.R. China.
Received 11 December 2011; Accepted (in revised version) 27 July 2012
Available online 15 November 2012
The local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrodinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrodinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis.AMS subject classifications: 65M06, 65M12, 65Z05, 70H15
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Key words: LOD-MS, Gross-Pitaevskii equation, local one-dimensional method, midpoint method, conservation laws.
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