TY - GEN

T1 - Numerical analysis of Schrödinger equations in the highly oscillatory regime

AU - Markowich, Peter A.

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Bloch decomposition

KW - Discretisation schemes

KW - Schrödinger equation

KW - Semiclassical asymptotics

KW - Spectral methods

KW - Wigner measure

UR - http://www.scopus.com/inward/record.url?scp=84877912439&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84877912439

SN - 9814324302

SN - 9789814324304

T3 - Proceedings of the International Congress of Mathematicians 2010, ICM 2010

SP - 2776

EP - 2804

BT - Proceedings of the International Congress of Mathematicians 2010, ICM 2010

T2 - International Congress of Mathematicians 2010, ICM 2010

Y2 - 19 August 2010 through 27 August 2010

ER -