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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bao, W. (2001). Time-splitting Chebyshev-spectral approximations for (non)linear Schrödinger equation, preprint.
Bao, W., Jin, Shi, and Markowich, P.A. (2001). On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., to appear.
Bao, W., Jin, Shi, and Markowich, P.A. (2001). Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semi-classical regimes, SIAM J. Sci. Comput., submitted.
Bronski, J.C., and McLaughlin, D.W., (1994). Semiclassical behavior in the NLS equation: optical shocks -focusing instabilities, Singular Limits of Dispersive Waves, Plenum Press, New York and London.
Ceniceros, H.D., and Tian, F.R. A numerical study of the semi-classical limit of the focusing nonlinear Schrödinger equation, Phys. Lett. A., to appear.
Gardiner, S.A., Jaksch, D., Dum, R., Cirac, J.I., and Zollar, P. (2000). Nonlinear matter wave dynamics with a chaotic potential, Phys. Rev. A, 62, pp. 023612–1:21.
Gasser, I., and Markowich, P. A., (1991).Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptotic Analysis 14, pp. 97–116.
Gerard, P., (1991). it Microlocal defect measures, Comm. PDE. 16, pp. 1761–1794.
Gerard, P., Markowich, P.A., Mauser, N.J., and Poupaud, F., (1997). Homogenization limits and Wigner transforms, Comm. Pure Appl. Math. 50, pp. 321–377.
Gottlieb, D., and Orszag, S.A., (1977). Numerical Analysis of Spectral Methods, SIAM, Philadelphia.
Grenier, E. (1998). Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc, 126, pp. 523–530.
Jin, Shan, Levermore, CD., and McLaughlin, D.W., (1999). The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math. LII, pp. 613–654.
Jin, Shan, Levermore, C.D., and McLaughlin, D.W., (1994). The behavior of solutions of the NLS equation in the semiclassical limit, Singular Limits of Dispersive Waves, Plenum Press, New York and London.
Jüngel, A. (2001). Quasi-hydrodynamic semiconductor equations, Progress in Nonlinear Differential Equations and Its Applications, Birkhäuser, Basel.
Krasny, R. (1986). A study of singularity formulation in a vortex sheet by the point-vortex approximation, J. Fluid Mech., 167, pp. 65–93.
Lin, C.K., and Li, H. (2001). Semiclassical limit and well-posedness of Schödinger-Poisson and quantum hydrodynamics, preprint.
Laudau, and Lifschitz (1977). Quantum Mechanics: non-relativistic theory, Pergamon Press, New York.
Markowich, P.A., Mauser, N.J., and Poupaud, F., (1994). A Wigner function approach to semiclassical limits: electrons in a periodic potential, J. Math. Phys. 35, pp. 1066–1094.
Markowich, P.A., Pietra, P., and Pohl, C., (1999). Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math. 81, pp. 595–630.
Markowich, P.A., Pietra, P., Pohl, C., and Stimming, H.P., (2000). A Wigner-Measure Analysis of the Dufort-Frankel scheme for the Schrödinger equation, preprint.
Miller, P.D., Kamvissis, S., (1998). On the semiclassical limit of the focusing nonlinear Schrödinger equation, Phys. Letters A, 247, pp. 75–86.
Tartar, L., (1990). H-measures: a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115, pp. 193–230.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media New York
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-1-4615-0113-8_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-4929-7
Online ISBN: 978-1-4615-0113-8
eBook Packages: Springer Book Archive