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Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation


Authors: Weizhu Bao and Yongyong Cai
Journal: Math. Comp. 82 (2013), 99-128
MSC (2010): Primary 35Q55, 65M06, 65M12, 65M22, 81-08
DOI: https://doi.org/10.1090/S0025-5718-2012-02617-2
Published electronically: June 20, 2012
MathSciNet review: 2983017
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Abstract: We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of $ O(h^2+\tau ^2)$ in the $ l^2$-norm and discrete $ H^1$-norm with time step $ \tau $ and mesh size $ h$. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain a priori bound of the numerical solution in the $ l^\infty $-norm by using the inverse inequality and the $ l^2$-norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods.


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  • 1. 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.
  • 2. S. K. Adhikari, Numerical study of the coupled time-dependent Gross-Pitaevskii equation: Application to Bose-Einstein condensation, Phy. Rev. E, 63 (2001), article 056704.
  • 3. G. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal., 13 (1993), pp. 115-124. MR 1199033 (94d:65046)
  • 4. G. Akrivis, V. Dougalis and O. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), pp. 31-53. MR 1103752 (92a:65256)
  • 5. M. H. Anderson, J. R. Ensher, M. R. Matthewa, C. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), pp. 198-201.
  • 6. W. Bao, Q. Du and Y. Zhang, Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation, SIAM J. Appl. Math., 66 (2006), pp. 758-786. MR 2216159 (2006k:35267)
  • 7. W. 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. MR 1977789 (2004g:82065)
  • 8. W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximation for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), pp. 487-524. MR 1880116
  • 9. W. Bao, H. Li and J. Shen, A generalized-Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates, SIAM J. Sci. Comput., 31 (2009), pp. 3685-3711. MR 2556558 (2010j:65193)
  • 10. W. Bao and J. Shen, A Fourth-order time-splitting Laguerre-Hermite pseudo-spectral method for Bose-Einstein condensates, SIAM J. Sci. Comput., 26 (2005), pp. 2010-2028. MR 2196586 (2006i:65177)
  • 11. W. Bao and H. Wang, An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates, J. Comput. Phys., 217 (2006), pp. 612-626. MR 2260616 (2007k:82132)
  • 12. W. Bao, H. Wang and P. A. Markowich, Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Commun. Math. Sci., 3 (2005), pp. 57-88. MR 2132826 (2006b:82013)
  • 13. C. Besse, B. Bidégaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), pp. 26-40. MR 1921908 (2003k:65099)
  • 14. B. M. Caradoc-Davis, R. J. Ballagh and K. Burnett, Coherent dynamics of vortex formation in trapped Bose-Einstein condensates, Phys. Rev. Lett., 83 (1999), pp. 895-898.
  • 15. T. Cazenave, Semilinear Schrödinger equations, (Courant Lecture Notes in Mathematics vol. 10), New York University, Courant Institute of Mathematical Sciences, AMS, 2003.
  • 16. Q. Chang, B. Guo and H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comp., 64 (1995), pp. 537-553. MR 1284664 (95f:65163)
  • 17. Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), pp. 397-415. MR 1669707 (99i:65086)
  • 18. K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), pp. 3969-3973.
  • 19. A. Debussche and E. Faou, Modified energy for split-step methods applied to the linear Schrödinger equations, SIAM J. Numer. Anal., 47 (2009), pp. 3705-3719. MR 2576517 (2010m:65169)
  • 20. C. M. Dion and E. Cances, Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap, Phys. Rev. E, 67 (2003), article 046706.
  • 21. R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp., 58 (1992), pp. 83-102. MR 1106968 (92e:65123)
  • 22. B. Y. Guo, The convergence of numerical method of nonlinear Schrödinger equation, J. Comput. Math., 4 (1986), 121-130. MR 854389 (89a:65136)
  • 23. C. Hao, L. Hsiao and H. Li, Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions, J. Math. Phys., 48 (2007), article 102105. MR 2362770 (2008m:35336)
  • 24. C. Hao, L. Hsiao and H. Li, Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term, Math. Meth. Appl. Sci., 31 (2008), pp. 655-664. MR 2400070 (2009f:35317)
  • 25. R. H. Hardin and F. D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Rev. Chronicle, 15 (1973), pp. 423.
  • 26. O. Karakashian, G. Akrivis and V. Dougalis, On optimal order error estimates for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 30 (1993), pp. 377-400. MR 1211396 (94c:65119)
  • 27. R. Landes, On Galerkin's method in the existence theory of quasilinear elliptic equations, J. Funct. Anal., 39 (1980), pp. 123-148. MR 597807 (83j:35064)
  • 28. M. Lees, Approximate solution of parabolic equations, J. Soc. Indust. Appl. Math., 7 (1959), pp. 167-183. MR 0110212 (22:1092)
  • 29. E. H. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Commun. Math. Phys., 264 (2006), pp. 505-537. MR 2215615 (2007h:82008)
  • 30. C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), pp. 2141-2153. MR 2429878 (2009d:65114)
  • 31. P. A. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math., 81 (1999), pp. 595-630. MR 1675220 (2000a:81038)
  • 32. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wiemann and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83 (1999), pp. 2498-2501.
  • 33. C. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential, BIT, 49 (2009), pp. 199-215. MR 2486135 (2009m:65163)
  • 34. K. Ohannes and M. Charalambos, A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), pp. 1779-1807. MR 1712169 (2000h:65139)
  • 35. L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003. MR 2012737 (2004i:82038)
  • 36. M. P. Robinson, G. Fairweather and B. M. Herbst, On the numerical solution of the cubic Schrödinger equation in one space variable, J. Comput. Phys., 104 (1993), pp. 277-284. MR 1198234
  • 37. T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations, II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys., 55 (1984), pp. 203-230. MR 762363 (86e:65128b)
  • 38. M. Thalhammer, High-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 46 (2008), pp. 2022-2038. MR 2399406 (2009b:65163)
  • 39. M. Thalhammer, M. Caliari and C. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), pp. 822-832. MR 2477790 (2010e:65179)
  • 40. V. Thomée, Galerkin finite element methods for parabolic problems, Springer, 1997. MR 1479170 (98m:65007)
  • 41. Y. L. Zhou, Implicit difference scheme for the generalized non-linear Schrödinger system, J. Comput. Math., 1 (1983), 116-129.

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Additional Information

Weizhu Bao
Affiliation: Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 119076
Email: bao@math.nus.edu.sg

Yongyong Cai
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119076
Email: caiyongyong@nus.edu.sg

DOI: https://doi.org/10.1090/S0025-5718-2012-02617-2
Keywords: Gross-Pitaevskii equation, angular momentum rotation, finite difference method, semi-implicit scheme, conservative Crank-Nicolson scheme
Received by editor(s): March 29, 2010
Received by editor(s) in revised form: March 26, 2011
Published electronically: June 20, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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