Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-27T23:01:03.671Z Has data issue: false hasContentIssue false
  • English
  • Français

LOD-MS for Gross-Pitaevskii Equation in Bose-Einstein Condensates

Published online by Cambridge University Press:  03 June 2015

Linghua Kong ,
Jialin Hong  and
Jingjing Zhang
Linghua Kong*
Affiliation:
School of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, P.R. China
Jialin Hong*
Affiliation:
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
Jingjing Zhang*
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, P.R. China
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

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 Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger 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.

Keywords

65M0665M1265Z0570H15LOD-MSGross-Pitaevskii equationlocal one-dimensional methodmidpoint methodconservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 1 , July 2013 , pp. 219 - 241
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aydin, A., Karasoözen, B., Lobatto IIIA-IIIB discretization of the strongly coupled nonlinear Schrödinger equation. J. Comput. Appl. Math., 235 (2011), pp. 4770–4779.Google Scholar
[2]Bao, W., Numerical methods for the nonlinear Schrödinger equation with nonzero far-field conditions, Methods and Applications Anal., 11 (2004), pp. 1–22.Google Scholar
[3]Bao, W., Du, Q., Computing the gounds state solution of Bose-Einstein Condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), pp. 1674–1697.CrossRefGoogle Scholar
[4]Bao, W., Du, Q., Zhang, Y.Z., Dynamics of rotating Bose-Einstein Condensates and its efficient and accurate numerical computation, SIAM J. Appl. Math., 66 (2006), pp. 758–786.Google Scholar
[5]Bridges, T.J., Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltoninan PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), pp. 184–193.CrossRefGoogle Scholar
[6]Cai, J., Wang, Y., Qiao, Z., Multisymplectic Preissman scheme for the time-domain Maxwell’s equations, J. Math. Phys., 50 (2009), pp. 033510.CrossRefGoogle Scholar
[7]Dalfovo, F., Giorgini, S., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), pp. 463–512.Google Scholar
[8]Du, Q., Ju, L., Numerical simulations of the quantized vortices on a thin superconducting hollow sphere, J. Comput. Phys., 201 (2004), pp. 511–530.CrossRefGoogle Scholar
[9]Gross, E.P., Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), pp. 195.Google Scholar
[10]Hairer, E., Lubich, C., Wanner, G., Geometric Numerical Integration Structure-preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer-Verlag, Berlin, 2006.Google Scholar
[11]Hong, J., Kong, L., Novel multi-symplectic integrators for nonlinear fourth-order Schrödinger equation with a trapped term, Commun. Comput. Phys., 7(2010), pp. 613–630.Google Scholar
[12]Hong, J., Liu, X., Li, C., Multi-symplectic Runge-Kutta-Nyström methods for nonlinear Schrödinger equations with variable coefficients, J. Comput. Phys., 226 (2007), pp. 1968–1984.Google Scholar
[13]Huang, H., Wang, L., Local one-dimensional multisymplectic integrator for Schröinger equation, J. Jiangxi Normal. Univer., 35 (2011), pp. 455–458.Google Scholar
[14]Islas, A.L., Schober, C.M., Multi-symplectic methods for generalized Schrödinger equations, Future Gener. Comput. Syst., 19 (2003), pp. 403–413.CrossRefGoogle Scholar
[15]Kong, L., Hong, J., Zhang, J., Splitting multi-symplectic methods for Maxwell’s equation, J. Comput. Phys., 229 (2010), pp. 4259–4278.CrossRefGoogle Scholar
[16]Morton, K.W., Mayers, D. F., Numerical solution of partial differential equations, Cambridges University Press: Cambridge, 2005.Google Scholar
[17]Peaceman, D., Rachford, H., The numerical solution of parabolic and elliptic equations, J. Soc. Indust. Appl. Math., 3 (1955), pp. 28–41.Google Scholar
[18]Pitaevskii, L.P., Vortex lines in an imperfect Bose gas, Zh. Eksp. Teor. Fiz. 40 (1961), pp. 646.Google Scholar
[19]Reich, S., Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equation, J. Comput. Phys., 157(2000), pp. 473–499.Google Scholar
[20]Ruprecht, P.A., Holland, M.J., Burrett, K., Edwards, M., Time-dependent solution of the nonlinear Schrödinger equation for Bose-condensed trapped neutral atoms, Phys. Rev. A, 51 (1995), pp. 4704–4711.Google Scholar
[21]Strang, G., On the construction and comparison of difference schemes. SIAMJ. Numer. Anal., 5 (1968), pp. 506–517.CrossRefGoogle Scholar
[22]Tappert, F., Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method. in: Newell, A.C.(Ed.), Nonlinear Wave Motion, Lect. Appl. Math., Amer. Math. Soc., Providence, RI, 15 (1974) 215–216.Google Scholar
[23]Tian, Y., M.Qin, , Zhang, Y., Ma, T., The multisymplectic numerical method for Gross-Pitaevskii equation, Comput. Phys. Commun., 178 (2008), pp. 449–458.CrossRefGoogle Scholar
[24]Wang, H., Numerical studies on split-step finite difference method for nonlinear Schrödinger equations, Appl. Math. Comput., 170 (2005), pp. 17–35.Google Scholar
[25]Wang, L., Multisymplectic Preissman scheme and its application. J. Jiangxi Normal Univer. 33 (2009), pp. 42–46.Google Scholar
[26]Zhang, J., Xu, Z., Wu, X., Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations: Two-dimensional case. Phys. Rev. E, 79 (2009), pp. 046711.Google Scholar
[27]Zhang, Y., Numerical study of vortex interactions in Bose-Einstein condensation. Commun. Comput. Phys., 8 (2010), pp. 327–350.Google Scholar