Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics

¹ Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Wien, Austria.
othmar@othmar-koch.org
² Institut für Mathematik, Leopold–Franzens Universität Innsbruck, Technikerstrasse 13/VII, 6020 Innsbruck, Austria.
christof.neuhauser@uibk.ac.at
³ Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr München, Werner–Heisenberg–Weg 39, 85577 Neubiberg, Germany.
mechthild.thalhammer@uibk.ac.at

Received: 10 August 2011
Revised: 14 February 2012

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.