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Exponential splitting for unbounded operators


Authors: Eskil Hansen and Alexander Ostermann
Journal: Math. Comp. 78 (2009), 1485-1496
MSC (2000): Primary 65M15, 65J10, 65L05, 35Q40
DOI: https://doi.org/10.1090/S0025-5718-09-02213-3
Published electronically: January 22, 2009
MathSciNet review: 2501059
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Abstract: We present a convergence analysis for exponential splitting methods applied to linear evolution equations. Our main result states that the classical order of the splitting method is retained in a setting of unbounded operators, without requiring any additional order condition. This is achieved by basing the analysis on the abstract framework of (semi)groups. The convergence analysis also includes generalizations to splittings consisting of more than two operators, and to variable time steps. We conclude by illustrating that the abstract results are applicable in the context of the Schrödinger equation with an external magnetic field or with an unbounded potential.


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Additional Information

Eskil Hansen
Affiliation: Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, A-6020 Innsbruck, Austria
Email: eskil.hansen@uibk.ac.at

Alexander Ostermann
Affiliation: Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, A-6020 Innsbruck, Austria
Email: alexander.ostermann@uibk.ac.at

DOI: https://doi.org/10.1090/S0025-5718-09-02213-3
Keywords: Exponential splitting, splitting schemes, convergence, nonstiff order, unbounded operators, Schr\"{o}dinger equation
Received by editor(s): February 29, 2008
Received by editor(s) in revised form: August 19, 2008
Published electronically: January 22, 2009
Additional Notes: This work was supported by the Austrian Science Fund under grant M961-N13.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.