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Numerical L Methods for Schrödinger Equations

In: Recent Progress in Computational and Applied PDES

Author

Listed:
  • Weizhu Bao

    (National University of Singapore, Department of Computational Science)

  • Shi Jin

    (University of Wisconsin-Madison, Department of Mathematics)

  • Peter A. Markowich

    (University of Vienna, Institute of Mathematics)

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.

Suggested Citation

  • Weizhu Bao & Shi Jin & Peter A. Markowich, 2002. "Numerical L Methods for Schrödinger Equations," Springer Books, in: Tony F. Chan & Yunqing Huang & Tao Tang & Jinchao Xu & Long-An Ying (ed.), Recent Progress in Computational and Applied PDES, pages 27-38, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-0113-8_2
    DOI: 10.1007/978-1-4615-0113-8_2
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