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Numerische Mathematik

, Volume 127, Issue 2, pp 315–364 | Cite as

Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross–Pitaevskii equations

  • Harald Hofstätter
  • Othmar Koch
  • Mechthild Thalhammer
Article

Abstract

A convergence analysis of time-splitting pseudo-spectral methods adapted for time-dependent Gross–Pitaevskii equations with additional rotation term is given. For the time integration high-order exponential operator splitting methods are studied, and the space discretization relies on the generalized-Laguerre–Fourier spectral method with respect to the \((x,y)\)-variables as well as the Hermite spectral method in the \(z\)-direction. Essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations, fractional power spaces defined by the principal linear part, a Sobolev-type inequality in a curved rectangle, and results on the asymptotical distribution of the nodes and weights associated with Gauß–Laguerre quadrature. The obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained, provided that the problem data satisfy suitable regularity requirements. A numerical example confirms the theoretical convergence estimate.

Mathematics Subject Classification (2000)

65L05 65M12 65J15 

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Dover Publications, New York (1992) (Reprint of the 1972 edition)Google Scholar
  2. 2.
    Adams, R.A.: Sobolev Spaces. Academic Press, Orlando (1975)zbMATHGoogle Scholar
  3. 3.
    Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Relat. Mod. 6, 1–135 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bao, W., Li, H., Shen, J.: A generalized-Laguerre–Fourier–Hermite pseudospectral method for computing the dynamics of rotating Bose–Einstein condensates. SIAM J. Sci. Comput. 31(5), 3685–3711 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bao, W., Shen, J.: A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose–Einstein condensates. SIAM J. Sci. Comput. 26(6), 2010–2028 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Blanes, S., Moan, P.C.: Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. J. Comput. Appl. Math. 142, 313–330 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Caliari, M., Neuhauser, Ch., Thalhammer, M.: High-order time-splitting Hermite and Fourier spectral methods for the Gross-Pitaevskii equation. J. Comput. Phys. 228, 822–832 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Danaila, I., Hecht, F.: A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose–Einstein condensates. J. Comput. Phys. 229, 6946–6960 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Danaila, I., Kazemi, P.: A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation. SIAM J. Sci. Comput. 32, 2447–2467 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gauckler, L.: Convergence of a split-step Hermite method for the Gross–Pitaevskii equation. IMA J. Numer. Anal. 31, 396–415 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, Oxford (2004)Google Scholar
  12. 12.
    Guo, B., Shen, J., Xu, Ch.: Spectral and pseudospectral approximations using Hermite functions: application to the Dirac equation. Adv. Comput. Math. 19, 35–55 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Koch, O., Neuhauser, Ch., Thalhammer, M.: Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics. M2AN Math. Model. Numer. Anal. 2012Google Scholar
  14. 14.
    Levin, E., Lubinsky, D.: Orthogonal polynomials for exponential weights \(x^{2\rho }e^{-2Q(x)}\) on \([0, d)\). J. Approx. Theory 134(2), 199–256 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Levin, E., Lubinsky, D.: Orthogonal polynomials for exponential weights \(x^{2\rho }e^{-2Q(x)}\) on \([0, d)\). II. J. Approx. Theory 139(1–2), 107–143 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lubich, C.: On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77, 2141–2153 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  18. 18.
    Remmert, R.: Theory of Complex Functions. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  19. 19.
    Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms. Analysis and Applications. Springer, Berlin (2011)CrossRefGoogle Scholar
  20. 20.
    Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46(4), 2022–2038 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Thalhammer, M.: Convergence analysis of high-order time-splitting pseudo-spectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50, 3231–3258 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Triebel, H.: Higher Analysis. Barth, Leipzig (1992)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Harald Hofstätter
    • 1
  • Othmar Koch
    • 1
  • Mechthild Thalhammer
    • 2
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria
  2. 2.Institut für Mathematik, Leopold-Franzens Universität InnsbruckInnsbruckAustria

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