133
Views
1
CrossRef citations to date
0
Altmetric
Original Articles

Optimal error estimates of explicit finite difference schemes for the coupled Gross–Pitaevskii equations

ORCID Icon &
Pages 1874-1892 | Received 27 Oct 2016, Accepted 09 Jun 2017, Published online: 03 Jul 2017

References

  • J.R. Abo-Shaeer, C. Raman, J.M. Vogels, and W. Ketterle, Observation of vortex lattices in Bose-Einstein condensates, Science 292 (2001), pp. 476–479. doi: 10.1126/science.1060182
  • S.K. Adhikari, Numerical solution of the two-dimensional Gross-Pitaevskii equation for trapped interacting atoms, Phys. Lett. A 265 (2000), pp. 91–96. doi: 10.1016/S0375-9601(99)00878-6
  • X. Antoine, W.Z. Bao, and C. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun. 184 (2013), pp. 2621–2633. doi: 10.1016/j.cpc.2013.07.012
  • W.Z. Bao, Ground states and dynamics of multicomponent Bose-Einstein condensates, Multiscale Model. Simul. 2 (2004), pp. 210–236. doi: 10.1137/030600209
  • W.Z. Bao and Y.Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models. 6 (2013), pp. 1–135. doi: 10.3934/krm.2013.6.1
  • W.Z. Bao and Y.Y. Cai, Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comput. 82 (2013), pp. 99–128. doi: 10.1090/S0025-5718-2012-02617-2
  • W.Z. Bao and D. Jaksch, An explicit unconditionally stable numerical methods for solving damped nonlinear Schrödinger equations with a focusing nonlinearity, SIAM J. Numer. Anal. 41 (2003), pp. 1406–1426. doi: 10.1137/S0036142902413391
  • W.Z. Bao and Y.Z. Zhang, Dynamics of the ground state and central vortex states in Bose-Einstein condensation, Math. Models Methods Appl. Sci. 15 (2005), pp. 1863–1896. doi: 10.1142/S021820250500100X
  • W.Z. Bao and Y.Z. Zhang, Dynamics of the ground state and central vortex states in Bose-Einstein condensation, Math. Models Methods Appl. Sci. 15 (2005), pp. 1863–1896. doi: 10.1142/S021820250500100X
  • W.Z. Bao, S. Jin, and P.A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys. 175 (2002), pp. 487–524. doi: 10.1006/jcph.2001.6956
  • W.Z. Bao, S. Jin, and P.A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput. 25 (2003), pp. 27–64. doi: 10.1137/S1064827501393253
  • W.Z. Bao, D. Jaksch, and P.A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys. 187 (2003), pp. 318–342. doi: 10.1016/S0021-9991(03)00102-5
  • W.Z. Bao, D. Marahrens, Q.L. Tang, and Y.Z. Zhang, A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via rotating Larangian coordinates, SIAM J. Sci. Comput. 35 (2013), pp. A2671–A2695. doi: 10.1137/130911111
  • M.M. Cerimele, M.L. Chiofalo, F. Pistella, S. Succi, and M.P. Tosi, Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: An application to trapped Bose–Einstein condensates, Phys. Rev. E 62 (2000), pp. 1382–1389. doi: 10.1103/PhysRevE.62.1382
  • S.-M. Chang, C.-S. Lin, T.-C. Lin, and W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Physica D 196 (2004), pp. 341–361. doi: 10.1016/j.physd.2004.06.002
  • S.-M. Chang, W.-W. Lin, and S.-F. Shieh, Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate, J. Comput. Phys. 202 (2005), pp. 367–390. doi: 10.1016/j.jcp.2004.07.012
  • A.L. Fetter and A.A. Svidzinsky, Vortices in a trapped dilute Bose-Einstein condensate, J. Phys. Condens. Matter. 13 (2001), pp. R135–R194. doi: 10.1088/0953-8984/13/12/201
  • Z. Gao and S.S. Xie, Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations, Appl. Numer. Math. 61 (2011), pp. 593–614. doi: 10.1016/j.apnum.2010.12.004
  • J.J. García-Ripoll, V.M. Pérez-García, and F. Sols, Split vortices in optically coupled Bose-Einstein condensates, Phys. Rev. A 66 (2002), p. 021602. doi: 10.1103/PhysRevA.66.021602
  • D.S. Hall, M.R. Matthews, J.R. Ensher, C.E. Wieman, and E.A. Cornell, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett. 81 (1998), pp. 1539–1542. doi: 10.1103/PhysRevLett.81.1539
  • X.L. Hu and L.M. Zhang, Conservative compact difference schemes for the coupled nonlinear Schrödinger system, Numer. Methods Partial Differ. Equ. 30 (2014), pp. 749–772. doi: 10.1002/num.21826
  • K. Kasamatsu, M. Tsubota, and M. Ueda, Vortices in multicomponent Bose-Einstein condensates, Internat. J. Modern Phys. B 19 (2005), pp. 1835–1904. doi: 10.1142/S0217979205029602
  • M.-C. Lai, C.-Y. Huang, and T.-S. Lin, A Simple Dufort-Frankel-type scheme for the Gross-Pitaevskii equation of Bose-Einstein condensates on different geometries, Numer. Methods Partial Differ. Equ. 20 (2004), pp. 624–638. doi: 10.1002/num.20008
  • H.-L. Liao, Z.-Z. Sun, and H.-S. Shi, Error estimate of fourth-order compact scheme for linear Schrödinger equations, SIAM J. Numer. Anal. 47 (2010), pp. 4381–4401. doi: 10.1137/080714907
  • J.F. Lu and Z. Guan, Numerical Solution for partial differential equation, 2nd ed., Tsinghua University Press, Beijing, 2004 (in Chinese).
  • K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett. 84 (2000), pp. 806–809. doi: 10.1103/PhysRevLett.84.806
  • M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, C.E. Wieman, and E.A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett. 83 (1999), pp. 201–202.
  • J. Ming, Q.L. Tang, and Y.Z. Zhang, An efficient spectral method for computing dynamics of rotating two-component Bose-Einstein condensates via coordinate transformation, J. Comput. Phys. 258 (2014), pp. 538–554. doi: 10.1016/j.jcp.2013.10.044
  • M. Modugno, F. Dalfovo, C. Fort, P. Maddaloni, and F. Minardi, Dynamics of two colliding Bose-Einstein condensates in an elongated magnetostatic trap, Phys. Rev. A 62 (2000), p. 063607.
  • F.I. Moxley.III, T. Byrnes, B.L. Ma, Y. Yan, and W.Z. Dai, A G-FDTD scheme for solving multi-dimensional open dissipative Gross-Pitaevskii equations, J. Comput. Phys. 282 (2015), pp. 303–316. doi: 10.1016/j.jcp.2014.11.021
  • P. Muruganandam and S.K. Adhikari, Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap, Comput. Phys. Commun. 180 (2009), pp. 1888–1912. doi: 10.1016/j.cpc.2009.04.015
  • C.J. Myatt, E.A. Burt, R.W. Ghrist, E.A. Cornell, and G.E. Wieman, Production of two overlapping Bose-Einstein condensates by sympathetic cooling, Phys. Rev. Lett. 78 (1997), pp. 586–589. doi: 10.1103/PhysRevLett.78.586
  • Z.Z. Sun, Numerical Solution for Partial Differential Equation, 2nd ed., Science Press, Beijing, 2012 (in Chinese).
  • Z.-Z. Sun and D.-D. Zhao, On the l∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations, Comput. Math. Appl. 59 (2010), pp. 3286–3300. doi: 10.1016/j.camwa.2010.03.012
  • T.C. Wang, Optimal point-wise error estimate of a compact difference scheme for the coupled Gross-Pitaevskii equations in one dimension, J. Sci. Comput. 41 (2014), pp. 158–186. doi: 10.1007/s10915-013-9757-1
  • T.C. Wang, Optimal point-wise error estimate of a compact finite difference scheme for the coupled nonlinear Schrödinger equations, J. Comput. Math. 32 (2014), pp. 58–74. doi: 10.4208/jcm.1310-m4340
  • T.C. Wang and X.F. Zhao, Optimal l∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions, Sci. China. Math. 57 (2014), pp. 2189–2214. doi: 10.1007/s11425-014-4773-7
  • T.C. Wang, T. Nie, L.M. Zhang, and F.Q. Chen, Numerical simulation of a nonlinearly coupled Schrödinger system: A linearly uncoupled finite difference scheme, Math. Comput. Simul. 79 (2008), pp. 607–621. doi: 10.1016/j.matcom.2008.03.017
  • T.C. Wang, T. Nie, and L.M. Zhang, Analysis of a symplectic difference scheme for a coupled nonlinear Schroödinger system, J. Comput. Appl. Math. 231 (2009), pp. 745–759. doi: 10.1016/j.cam.2009.04.022
  • J.E. Williams and M.J. Holland, Preparing topological states of a Bose-Einstein condensate, Nature 401 (1999), pp. 568–572. doi: 10.1038/44095
  • Y.Q. Xu and L.M. Zhang, Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation, Comput. Phys. Commun. 183 (2012), pp. 1082–1093. doi: 10.1016/j.cpc.2012.01.006
  • Y.L. Zhou, Application of Discrete Functional Analysis to the Finite Difference Methods, International Academic Publishers, Beijing, 1990.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.