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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.
@article{M2AN_2013__47_5_1265_0,
author = {Koch, Othmar and Neuhauser, Christof and Thalhammer, Mechthild},
title = {Error analysis of high-order splitting methods for nonlinear evolutionary {Schr\"odinger} equations and application to the {MCTDHF} equations in electron dynamics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1265--1286},
publisher = {EDP-Sciences},
volume = {47},
number = {5},
year = {2013},
doi = {10.1051/m2an/2013067},
mrnumber = {3100763},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013067/}
}
TY - JOUR AU - Koch, Othmar AU - Neuhauser, Christof AU - Thalhammer, Mechthild TI - Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1265 EP - 1286 VL - 47 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013067/ DO - 10.1051/m2an/2013067 LA - en ID - M2AN_2013__47_5_1265_0 ER -
%0 Journal Article %A Koch, Othmar %A Neuhauser, Christof %A Thalhammer, Mechthild %T Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1265-1286 %V 47 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013067/ %R 10.1051/m2an/2013067 %G en %F M2AN_2013__47_5_1265_0
Koch, Othmar; Neuhauser, Christof; Thalhammer, Mechthild. Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1265-1286. doi: 10.1051/m2an/2013067
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