On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
Bao, W. Jin, S. Markowich, P.A.
Jin, S.
Markowich, P.A.
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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 e is small. In this regime, the equation propagates oscillations with a wavelength of O (e), and finite difference approximations require the spatial mesh size h = o (e) and the time step k = o (e) 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 (e) and h = O (e). 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 e, and h = O (e)) 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. © 2002 Elsevier Science (USA).
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Journal of Computational Physics
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Date
2002-01-20
DOI
10.1006/jcph.2001.6956
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Article