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Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schrödinger equations

Authors: Philippe Chartier, Florian Méhats, Mechthild Thalhammer and Yong Zhang
Journal: Math. Comp. 85 (2016), 2863-2885
MSC (2010): Primary 34K33, 37L05, 35Q55
Published electronically: February 16, 2016
MathSciNet review: 3522973
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Abstract: In this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schrödinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang splitting method applied to the cubic Schrödinger equation is adapted to a wider class of nonlinearities. In a second step, the dependence of the global error on the decisive parameter $ 0 < \varepsilon <\!\!< 1$, defining the length of the period, is examined. The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor  $ \varepsilon $, provided that the time stepsize is chosen as an integer fraction of the period. This improved error behavior over a time interval of fixed length, which is independent of the period, is due to an averaging effect. The extension of the convergence result to higher-order splitting methods and numerical illustrations complement the investigations.

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Additional Information

Philippe Chartier
Affiliation: INRIA-Rennes, IRMAR, ENS Cachan Bretagne, IPSO Project Team, Campus de Beaulieu, 35042 Rennes Cedex, France.

Florian Méhats
Affiliation: IRMAR, Université de Rennes 1, INRIA-Rennes, IPSO Project Team, Campus de Beaulieu, 35042 Rennes Cedex, France.

Mechthild Thalhammer
Affiliation: Leopold-Franzens Universität Innsbruck, Institut für Mathematik, Technikerstraße 13/VII, 6020 Innsbruck, Austria.

Yong Zhang
Affiliation: Wolfgang Pauli Institut c/o Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Keywords: Highly-oscillatory differential equations, time-dependent nonlinear Schr\"odinger equations, operator splitting methods, Strang splitting method, error estimate, convergence result, averaging
Received by editor(s): March 7, 2014
Received by editor(s) in revised form: March 30, 2015
Published electronically: February 16, 2016
Additional Notes: Corresponding author: Philippe Chartier
The authors acknowledge financial support by the Agence nationale de la recherche (ANR) within the project LODIQUAS ANR-11-IS01-0003 and the project Moonrise ANR-14-CE23-0007-01, by the Austrian Science Fund (FWF) under SFP Vienna Computational Materials Laboratory (ViCoM) and project P21620-N13, and by the Austrian Ministry of Science and Research via its grant for the WPI. The presented numerical results have been achieved by using the Vienna Scientific Cluster.
Article copyright: © Copyright 2016 American Mathematical Society

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