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Error analysis of high-order splitting methodsfor nonlinear evolutionary Schrödinger equationsand application to the MCTDHF equationsin electron dynamics

Published online by Cambridge University Press:  09 July 2013

Othmar Koch ,
Christof Neuhauser  and
Mechthild Thalhammer
Othmar Koch
Affiliation:
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Wien, Austria.. othmar@othmar-koch.org
Christof Neuhauser
Affiliation:
Institut für Mathematik, Leopold–Franzens Universität Innsbruck, Technikerstrasse 13/VII, 6020 Innsbruck, Austria.; christof.neuhauser@uibk.ac.at
Mechthild Thalhammer
Affiliation:
Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr München, Werner–Heisenberg–Weg 39, 85577 Neubiberg, Germany.; mechthild.thalhammer@uibk.ac.at
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Abstract

In this work, the error behaviour of high-order exponential operator splitting methods for the time integration of nonlinear evolutionary Schrödinger equations is investigated. The theoretical analysis utilises the framework of abstract evolution equations on Banach spaces and the formal calculus of Lie derivatives. The general approach is substantiated on the basis of a convergence result for exponential operator splitting methods of (nonstiff) order p applied to the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations, which are associated with a model reduction for high-dimensional linear Schrödinger equations describing free electrons that interact by Coulomb force. Provided that the analytical solution of the MCTDHF equations constituting a system of coupled linear ordinary differential equations and low-dimensional nonlinear partial differential equations satisfies suitable regularity requirements, convergence of order p − 1 in the H1 Sobolev norm and convergence of order p in the L2 norm is proven. An analogous result follows for the cubic nonlinear Schrödinger equation, which is also illustrated by a numerical experiment.

Keywords

Nonlinear evolution equationstime-dependent nonlinear Schrödinger equationsmulti-configuration time-dependent Hartree–Fock (MCTDHF) equationshigh-order exponential operator splitting methodslocal error expansionconvergence
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1265 - 1286
Copyright
© EDP Sciences, SMAI, 2013

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