Blanes Zamora, S.; Casas, F.; González, C.; Thalhammer, M. (2021). Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrodinger equations. IMA Journal of Numerical Analysis. 41(1):594-617. https://doi.org/10.1093/imanum/drz058
Por favor, use este identificador para citar o enlazar este ítem: http://hdl.handle.net/10251/181793
Title:
|
Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrodinger equations
|
Author:
|
Blanes Zamora, Sergio
Casas, Fernando
González, Cesáreo
Thalhammer, Mechthild
|
UPV Unit:
|
Universitat Politècnica de València. Departamento de Matemática Aplicada - Departament de Matemàtica Aplicada
|
Issued date:
|
|
Abstract:
|
[EN] 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 Schrodinger equations; a detailed stability and ...[+]
[EN] 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 Schrodinger 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 Schrodinger 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-ofconstants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.
[-]
|
Subjects:
|
Nonautonomous linear evolution equations
,
Schrödinger equations
,
Quantum systems
,
Time
integration methods
,
Exponential integrators
,
Magnus integrators
,
Commutator-free quasi-Magnus exponential integrators
,
Stability
,
Local error
,
Convergence
|
Copyrigths:
|
Reserva de todos los derechos
|
Source:
|
IMA Journal of Numerical Analysis. (issn:
0272-4979
)
|
DOI:
|
10.1093/imanum/drz058
|
Publisher:
|
Oxford University Press
|
Publisher version:
|
https://doi.org/10.1093/imanum/drz058
|
Project ID:
|
info:eu-repo/grantAgreement/MINECO//MTM2016-77660-P//NUEVOS RETOS EN INTEGRACION NUMERICA: FUNDAMENTOS ALGEBRAICOS, METODOS DE ESCISION, METODOS DE MONTECARLO Y OTRAS APLICACIONES/
|
Thanks:
|
Ministerio de Economia y Competitividad (Spain) (project MTM2016-77660-P (AEI/FEDER, UE) to S.B., F.C. and C.G.).
|
Type:
|
Artículo
|