On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime

  • Authors:

  • Weizhu Bao;Shi Jin;Peter A. Markowich

  • Affiliations:

  • Department of Computational Science, National University of Singapore 117543;Department of Mathematics, University of Wisconsin-Madison, Madison Wisconsin 53706;Institute of Mathematics, University of Vienna Boltzmanngasse 9, A-1090 Vienna, Austria

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2002

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Abstract

In this paper we study time-splitting spectral approximations for the linear Schrödinger equation in the semiclassical regime, where the Planck constant ε is small. In this regime, the equation propagates oscillations with a wavelength of O (ε), and finite difference approximations require the spatial mesh size h = o(ε) and the time step k = o(ε) in order to obtain physically correct observables. Much sharper mesh-size constraints are necessary for a uniform L2-approximation of the wave function. The spectral time-splitting approximation under study will be proved to be unconditionally stable, time reversible, and gauge invariant. It conserves the position density and gives uniform L2-approximation of the wave function for k = o(ε) and h = O(ε). Extensive numerical examples in both one and two space dimensions and analytical considerations based on the Wigner transform even show that weaker constraints (e.g., k independent of ε, and h = O (ε)) are admissible for obtaining "correct" observables. Finally, we address the application to nonlinear Schrödinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies.