Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations
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Altres documents de l'autoria: Blanes, Sergio; Casas, Fernando; González, Cesáreo; Thalhammer, Mechthild
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Títol
Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equationsData de publicació
2020-03-02Editor
Oxford University Press; Institute of Mathematics and its ApplicationsISSN
0272-4979; 1464-3642Cita bibliogràfica
Sergio Blanes, Fernando Casas, Cesáreo González, Mechthild Thalhammer, Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations, IMA Journal of Numerical Analysis, Volume 41, Issue 1, January 2021, Pages 594–617, https://doi.org/10.1093/imanum/drz058Tipus de document
info:eu-repo/semantics/articleVersió
info:eu-repo/semantics/acceptedVersionParaules clau / Matèries
Resum
This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrödinger equations; a detailed stability and ... [+]
This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrödinger equations; a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix; this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge–Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved; this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker–Campbell–Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrödinger equations is related to an appropriate treatment of compositions of evolution operators; an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-of-constants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data. [-]
Publicat a
IMA Journal of Numerical Analysis, Volume 41, Issue 1, January 2021, Pages 594–617, https://doi.org/10.1093/imanum/drz058Entitat finançadora
Ministerio de Economía y Competitividad (Spain)
Codi del projecte o subvenció
MTM2016-77660-P
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© The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
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