A simple Dufort-Frankel-type scheme for the Gross-Pitaevskii equation of Bose-Einstein condensates on different geometries

Numerical Methods for Partial Differential Equations

Volumn 20, Issue 4, 2004, Pages 624-638

  Lai, Ming Chih ^a   Huang, Chung Yin ^b   Lin, Te Sheng ^b  

^a NATIONAL CHIAO TUNG UNIVERSITY   (Taiwan)

^b NATIONAL CHUNG CHENG UNIVERSITY   (Taiwan)

Author keywords

Bose Einstein condensates; Dufort Frankel scheme; Gross Pitaevskii equation; Nonlinear Schr dinger equation

Indexed keywords

EID: 2942750508     PISSN: 0749159X     EISSN: None     Source Type: Journal    
DOI: 10.1002/num.20008     Document Type: Article

References (21)

1. 1 M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose‐Einstein condensation in a dilute atomic vapor, Science 269 (1995), 198–201.
2. 2 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 78 (1995), 3969–3973.
3. 3 F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose‐Einstein condensation in trapped gases, Rev Mod Phys 71 (3) (1999), 463–512.
4. 4 S. K. Adhikari, Numerical study of the spherically symmetric Gross‐Pitaevskii equation in two space dimensions, Phys Rev E 62 (2) (2000), 2937–2944.
5. 5 W. Bao, D. Jaksch, and P. A. Markowich, Numerical solution of the Gross‐Pitaevskii equation for Bose‐Einstein condensation, J Comput Phys 187 (2003), 318–342.
6. 6 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 (1) (2000), 1382–1389.
7. 7 M. M. Cerimele, F. Pistella, and S. Succi, Particle‐inspired scheme for the Gross‐Pitaevskii equation: an application to Bose‐Einstein condensation, Comput Phys Commun 129 (2000), 82–90.
8. 8 Q. Chang, E. Jia, and W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J Comput Phys 148 (1999), 397–415.
9. 9 E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento 20 (1961), 454–477.
10. 10 L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov Phys JETP 13 (1961), 451–454.
11. 11 J. C. Strikwerda, Finite difference schemes and partial differential equations, Wadsworth & Brooks/Cole, 1989.
12. 12 L. Wu, Dufort‐Frankel‐type methods for linear and nonlinear Schrödinger equations, SIAM J Numer Anal 33 (4) (1996), 1526–1533.
13. 13 F. Ivanauskas and M. Radžiūnas, On convergence and stability of the explicit difference method for solution of nonlinear Schrödinger equations, SIAM J Numer Anal 36 (5) (1999), 1466–1481.
14. 14 P. A. Markowich, P. Pietra, C. Pohl, and H. P. Stimming, A Wigner‐measure analysis of the Dufort‐Frankel scheme for the Schrödinger equation, SIAM J Numer Anal 40 (4) (2002), 1281–1310.
15. 15 P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Time‐dependent solution of the nonlinear Schrödinger equation for Bose‐condensed trapped neutral atoms, Phys Rev A 51 (1995), 4704–4711.
16. 16 W. Bao, S. Jin, and P. A. Markowich, On time‐splitting spectral approximation for the Schrödinger equations in the semi‐classical regime, J Comput Phys 175 (2002), 487–524.
17. 17 W. Bao, S. Jin, and P. A. Markowich, Numerical study of time‐splitting spectral discretizations of nonlinear Schrödinger equations in the semi‐classical regimes, SIAM J Sci Comput 25 (1) (2003), 27–64.
18. 18 W. Bao and J. Shen, A fourth‐order time‐splitting Laguerre‐Hermite pseudo‐spectral method for Bose‐Einstein condensates, preprint, 2003.
19. 19 M.‐C. Lai and W.‐C. Wang, Fast direct solvers for Poisson equation on polar and spherical geometries, Numer Methods Partial Differential Eq 18 (2002), 56–68.
20. 20 W. Bao and Q. Du, Computing the ground state solution of Bose‐Einstein condensates by a normalized gradient flow, preprint, 2003.
21. 21 M. J. Holland, D. S. Jin, M. L. Chiofalo, and J. Cooper, Emergence of interaction effects in Bose‐Einstein condensation, Phys Rev Lett 78 (1997), 3801–3805.