Numerical analysis of Schrödinger equations in the highly oscillatory regime

Peter Markowich*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Linear (and nonlinear) Schrödinger equations in the semiclassical (small dispersion) regime pose a significant challenge to numerical analysis and scientific computing, mainly due to the fact that they propagate high frequency spatial and temporal oscillations. At first we prove using Wigner measure techniques that finite difference discretisations in general require a disproportionate amount of computational resources, since underlying numerical meshes need to be fine enough to resolve all oscillations of the solution accurately, even if only accurate observables are required. This can be mitigated by using a spectral (in space) discretisation, combined with appropriate time splitting. Such discretisations are time-transverse invariant and allow for much coarser meshes than finite difference discretisations. In many physical applications highly oscillatory periodic potentials occur in Schrödinger equations, still aggrevating the oscillatory solution structure. For such problems we present a numerical method based on the Bloch decomposition of the wave function.

Original languageEnglish (US)
Title of host publicationProceedings of the International Congress of Mathematicians 2010, ICM 2010
Pages2776-2804
Number of pages29
StatePublished - Dec 1 2010
EventInternational Congress of Mathematicians 2010, ICM 2010 - Hyderabad, India
Duration: Aug 19 2010Aug 27 2010

Publication series

NameProceedings of the International Congress of Mathematicians 2010, ICM 2010

Other

OtherInternational Congress of Mathematicians 2010, ICM 2010
CountryIndia
CityHyderabad
Period08/19/1008/27/10

Keywords

  • Bloch decomposition
  • Discretisation schemes
  • Schrödinger equation
  • Semiclassical asymptotics
  • Spectral methods
  • Wigner measure

ASJC Scopus subject areas

  • Mathematics(all)

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