Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation
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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.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.
- 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.
- Georgios D. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal. 13 (1993), no. 1, 115–124. MR 1199033, DOI 10.1093/imanum/13.1.115
- Georgios D. Akrivis, Vassilios A. Dougalis, and Ohannes A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math. 59 (1991), no. 1, 31–53. MR 1103752, DOI 10.1007/BF01385769
- 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.
- Weizhu Bao, Qiang Du, and Yanzhi Zhang, Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation, SIAM J. Appl. Math. 66 (2006), no. 3, 758–786. MR 2216159, DOI 10.1137/050629392
- Weizhu Bao, Dieter Jaksch, and Peter A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys. 187 (2003), no. 1, 318–342. MR 1977789, DOI 10.1016/S0021-9991(03)00102-5
- Weizhu Bao, Shi Jin, and Peter A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys. 175 (2002), no. 2, 487–524. MR 1880116, DOI 10.1006/jcph.2001.6956
- Weizhu Bao, Hailiang Li, and Jie Shen, A generalized Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates, SIAM J. Sci. Comput. 31 (2009), no. 5, 3685–3711. MR 2556558, DOI 10.1137/080739811
- Weizhu Bao and Jie Shen, A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates, SIAM J. Sci. Comput. 26 (2005), no. 6, 2010–2028. MR 2196586, DOI 10.1137/030601211
- Weizhu Bao and Hanquan Wang, An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates, J. Comput. Phys. 217 (2006), no. 2, 612–626. MR 2260616, DOI 10.1016/j.jcp.2006.01.020
- Weizhu Bao, Hanquan Wang, and Peter A. Markowich, Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Commun. Math. Sci. 3 (2005), no. 1, 57–88. MR 2132826
- Christophe Besse, Brigitte Bidégaray, and Stéphane Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 40 (2002), no. 1, 26–40. MR 1921908, DOI 10.1137/S0036142900381497
- 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.
- T. Cazenave, Semilinear Schrödinger equations, (Courant Lecture Notes in Mathematics vol. 10), New York University, Courant Institute of Mathematical Sciences, AMS, 2003.
- Qian Shun Chang, Bo Ling Guo, and Hong Jiang, Finite difference method for generalized Zakharov equations, Math. Comp. 64 (1995), no. 210, 537–553, S7–S11. MR 1284664, DOI 10.1090/S0025-5718-1995-1284664-5
- Qianshun Chang, Erhui Jia, and W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys. 148 (1999), no. 2, 397–415. MR 1669707, DOI 10.1006/jcph.1998.6120
- 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.
- Arnaud Debussche and Erwan Faou, Modified energy for split-step methods applied to the linear Schrödinger equation, SIAM J. Numer. Anal. 47 (2009), no. 5, 3705–3719. MR 2576517, DOI 10.1137/080744578
- 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.
- R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp. 58 (1992), no. 197, 83–102. MR 1106968, DOI 10.1090/S0025-5718-1992-1106968-6
- Ben Yu Guo, The convergence of numerical method for nonlinear Schrödinger equation, J. Comput. Math. 4 (1986), no. 2, 121–130. MR 854389
- Chengchun Hao, Ling Hsiao, and Hai-Liang Li, Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions, J. Math. Phys. 48 (2007), no. 10, 102105, 11. MR 2362770, DOI 10.1063/1.2795218
- Chengchun Hao, Ling Hsiao, and Hai-Liang Li, Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term, Math. Methods Appl. Sci. 31 (2008), no. 6, 655–664. MR 2400070, DOI 10.1002/mma.931
- 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.
- Ohannes Karakashian, Georgios D. Akrivis, and Vassilios A. Dougalis, On optimal order error estimates for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 30 (1993), no. 2, 377–400. MR 1211396, DOI 10.1137/0730018
- R. Landes, On Galerkin’s method in the existence theory of quasilinear elliptic equations, J. Functional Analysis 39 (1980), no. 2, 123–148. MR 597807, DOI 10.1016/0022-1236(80)90009-9
- Milton Lees, Approximate solutions of parabolic equations, J. Soc. Indust. Appl. Math. 7 (1959), 167–183. MR 110212
- Elliott H. Lieb and Robert Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Comm. Math. Phys. 264 (2006), no. 2, 505–537. MR 2215615, DOI 10.1007/s00220-006-1524-9
- Christian Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp. 77 (2008), no. 264, 2141–2153. MR 2429878, DOI 10.1090/S0025-5718-08-02101-7
- Peter A. Markowich, Paola Pietra, and Carsten Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math. 81 (1999), no. 4, 595–630. MR 1675220, DOI 10.1007/s002110050406
- 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.
- Christof Neuhauser and Mechthild Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential, BIT 49 (2009), no. 1, 199–215. MR 2486135, DOI 10.1007/s10543-009-0215-2
- Ohannes Karakashian and Charalambos Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal. 36 (1999), no. 6, 1779–1807. MR 1712169, DOI 10.1137/S0036142997330111
- Lev Pitaevskii and Sandro Stringari, Bose-Einstein condensation, International Series of Monographs on Physics, vol. 116, The Clarendon Press, Oxford University Press, Oxford, 2003. MR 2012737
- 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), no. 1, 277–284. MR 1198234, DOI 10.1006/jcph.1993.1029
- Thiab R. Taha and Mark J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys. 55 (1984), no. 2, 203–230. MR 762363, DOI 10.1016/0021-9991(84)90003-2
- Mechthild Thalhammer, High-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal. 46 (2008), no. 4, 2022–2038. MR 2399406, DOI 10.1137/060674636
- Mechthild Thalhammer, Marco Caliari, and Christof Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys. 228 (2009), no. 3, 822–832. MR 2477790, DOI 10.1016/j.jcp.2008.10.008
- Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170, DOI 10.1007/978-3-662-03359-3
- Y. L. Zhou, Implicit difference scheme for the generalized non-linear Schrödinger system, J. Comput. Math., 1 (1983), 116-129.
Additional Information
- Weizhu Bao
- Affiliation: Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 119076
- MR Author ID: 354327
- Email: bao@math.nus.edu.sg
- Yongyong Cai
- Affiliation: Department of Mathematics, National University of Singapore, Singapore 119076
- Email: caiyongyong@nus.edu.sg
- Received by editor(s): March 29, 2010
- Received by editor(s) in revised form: March 26, 2011
- Published electronically: June 20, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2983017