Uniformly exponentially stable approximations for a class of damped systems

We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be l...

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Detalles Bibliográficos
Autores: Ervedoza, S., Zuazua, E.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2009
País:España
Institución:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/504
Acceso en línea:http://hdl.handle.net/20.500.11824/504
Access Level:acceso abierto
Palabra clave:Damped systems
Stabilization
Time semi-discretizations
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spelling Uniformly exponentially stable approximations for a class of damped systemsErvedoza, S.Zuazua, E.Damped systemsStabilizationTime semi-discretizationsWe consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi-discretizations to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a suitable CFL type condition on the time and space discretization parameters. © 2008 Elsevier Masson SAS. All rights reserved.201720172009info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/504reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttps://www.scopus.com/inward/record.uri?eid=2-s2.0-57849116170&doi=10.1016%2fj.matpur.2008.09.002&partnerID=40&md5=c1b83981065b63fdec3d03857d5eb908Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/5042026-06-19T12:47:47Z
dc.title.none.fl_str_mv Uniformly exponentially stable approximations for a class of damped systems
title Uniformly exponentially stable approximations for a class of damped systems
spellingShingle Uniformly exponentially stable approximations for a class of damped systems
Ervedoza, S.
Damped systems
Stabilization
Time semi-discretizations
title_short Uniformly exponentially stable approximations for a class of damped systems
title_full Uniformly exponentially stable approximations for a class of damped systems
title_fullStr Uniformly exponentially stable approximations for a class of damped systems
title_full_unstemmed Uniformly exponentially stable approximations for a class of damped systems
title_sort Uniformly exponentially stable approximations for a class of damped systems
dc.creator.none.fl_str_mv Ervedoza, S.
Zuazua, E.
author Ervedoza, S.
author_facet Ervedoza, S.
Zuazua, E.
author_role author
author2 Zuazua, E.
author2_role author
dc.subject.none.fl_str_mv Damped systems
Stabilization
Time semi-discretizations
topic Damped systems
Stabilization
Time semi-discretizations
description We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi-discretizations to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a suitable CFL type condition on the time and space discretization parameters. © 2008 Elsevier Masson SAS. All rights reserved.
publishDate 2009
dc.date.none.fl_str_mv 2009
2017
2017
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dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/504
url http://hdl.handle.net/20.500.11824/504
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://www.scopus.com/inward/record.uri?eid=2-s2.0-57849116170&doi=10.1016%2fj.matpur.2008.09.002&partnerID=40&md5=c1b83981065b63fdec3d03857d5eb908
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
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