The aim of this thesis was to elaborate and extend the results proved in the paper "Convergence of a split-step Hermite method for the Gross-Pitaevskii equation" by L. Gauckler (2010). There, one can find convergence analyses for time and space semi-discretisations and full discretisations applied to the cubic nonlinear Schrödinger equation with a harmonic oscillator potential. The methods used in this treatise include Hermite quadrature and an operator splitting of second order.<br />The author was able to generalise all these results for Schrödinger equations with a scaled harmonic oscillator potential and a sum of power-nonlinearities up to an arbitrary degree, and was also capable of showing an existence and uniqueness result for equations of this type.<br />Furthermore, using additionally the formal calculus of Lie derivatives, the author could prove convergence of arbitrary order of the time semi-discretised equation when using an appropriate higher order splitting scheme.<br />Proving a higher rate of convergence for the fully discretised scheme, however, turned out to be impossible without additional tools, and remains a challenge for future studies.
en
dc.language
English
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dc.language.iso
en
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dc.rights.uri
http://rightsstatements.org/vocab/InC/1.0/
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dc.subject
Schrödinger Gleichung
de
dc.subject
Hermite-Quadratur
de
dc.subject
Lie-Ableitungen
de
dc.subject
Operator-Splitting
de
dc.subject
Schrödinger equation
en
dc.subject
Hermite-quadrature
en
dc.subject
Lie derivatives
en
dc.subject
operator-splitting
en
dc.title
Convergence analysis of time-splitting pseudo-Hermite collocation methods applied to nonlinear Schrödinger equations
en
dc.type
Thesis
en
dc.type
Hochschulschrift
de
dc.rights.license
In Copyright
en
dc.rights.license
Urheberrechtsschutz
de
dc.contributor.affiliation
TU Wien, Österreich
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dc.rights.holder
Nicola Ondracek
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tuw.version
vor
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tuw.thesisinformation
Technische Universität Wien
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tuw.publication.orgunit
E101 - Institut für Analysis und Scientific Computing