Mathematical and computational methods for semiclassical Schrödinger equations

Handle URI:
http://hdl.handle.net/10754/598766
Title:
Mathematical and computational methods for semiclassical Schrödinger equations
Authors:
Jin, Shi; Markowich, Peter; Sparber, Christof
Abstract:
We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime. © 2011 Cambridge University Press.
Citation:
Jin S, Markowich P, Sparber C (2011) Mathematical and computational methods for semiclassical Schrödinger equations. Acta Numerica 20: 121–209. Available: http://dx.doi.org/10.1017/s0962492911000031.
Publisher:
Cambridge University Press (CUP)
Journal:
Acta Numerica
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
28-Apr-2011
DOI:
10.1017/s0962492911000031
Type:
Article
ISSN:
0962-4929; 1474-0508
Sponsors:
Partially supported by NSF grant no. DMS-0608720, NSF FRG grant DMS-0757285,a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the Universityof Wisconsin–Madison.Supported by a Royal Society Wolfson Research Merit Award and by KAUST througha Investigator Award KUK-I1-007-43.Partially supported by the Royal Society through a University Research Fellowship.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorJin, Shien
dc.contributor.authorMarkowich, Peteren
dc.contributor.authorSparber, Christofen
dc.date.accessioned2016-02-25T13:40:47Zen
dc.date.available2016-02-25T13:40:47Zen
dc.date.issued2011-04-28en
dc.identifier.citationJin S, Markowich P, Sparber C (2011) Mathematical and computational methods for semiclassical Schrödinger equations. Acta Numerica 20: 121–209. Available: http://dx.doi.org/10.1017/s0962492911000031.en
dc.identifier.issn0962-4929en
dc.identifier.issn1474-0508en
dc.identifier.doi10.1017/s0962492911000031en
dc.identifier.urihttp://hdl.handle.net/10754/598766en
dc.description.abstractWe consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime. © 2011 Cambridge University Press.en
dc.description.sponsorshipPartially supported by NSF grant no. DMS-0608720, NSF FRG grant DMS-0757285,a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the Universityof Wisconsin–Madison.Supported by a Royal Society Wolfson Research Merit Award and by KAUST througha Investigator Award KUK-I1-007-43.Partially supported by the Royal Society through a University Research Fellowship.en
dc.publisherCambridge University Press (CUP)en
dc.titleMathematical and computational methods for semiclassical Schrödinger equationsen
dc.typeArticleen
dc.identifier.journalActa Numericaen
dc.contributor.institutionUniversity of Wisconsin Madison, Madison, United Statesen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionUniversity of Illinois at Chicago, Chicago, United Statesen
kaust.grant.numberKUK-I1-007-43en
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