Abstract
In this note we review the time-splitting spectral method, recently studied by the authors, for linear[2] and nonlinear[3] Schrödinger equations (NLS) in the semiclassical regimes, where the Planck constant ɛ is small. The time-splitting spectral method under study is unconditionally stable and conserves the position density. Moreover it is gauge invariant and time reversible when the corresponding Schrödinger equation is. Numerical tests are presented for linear, for weak/strong focusing/defocusing nonlinearities, for the Gross-Pitaevskii equation and for current-relaxed quantum hydrodynamics. The tests are geared towards understanding admissible meshing strategies for obtaining ‘correct’ physical observables in the semi-classical regimes. Furthermore, comparisons between the solutions of the nonlinear Schrödinger equation and its hydrodynamic semiclassical limit are presented.
This research was supported by the International Erwin Schrödingcr Institute in Vienna. W.B. acknowledges support in part by the National University of Singapore gram No. R-151 -000-016-112. S.J. acknowledges support in pan by NSF grant No. DMS-0I96I06. P.A.M. acknowledges support from the EU-funded TMR network ‘Asymptotic Methods in kinetic Theory’ and from his WITTGENSTEIN-AWARD 2000. funded by the Austrian National Science Fund FWF.
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Bao, W., Jin, S., Markowich, P.A. (2002). Numerical L Methods for Schrödinger Equations. In: Chan, T.F., Huang, Y., Tang, T., Xu, J., Ying, LA. (eds) Recent Progress in Computational and Applied PDES. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0113-8_2
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DOI: https://doi.org/10.1007/978-1-4615-0113-8_2
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