Time-average on the numerical integration of nonautonomous differential equations

[EN] In this work we show how to numerically integrate nonautonomous differential equations by solving alternate time-averaged differential equations. Given a quadrature rule of order 2s or higher for s = 1, 2, . . . , we show how to build a differential equation with an averaged vector field that i...

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Autor: Blanes Zamora, Sergio|||0000-0001-5819-8898
Tipo de recurso: artículo
Fecha de publicación:2018
País:España
Institución:Universitat Politècnica de València (UPV)
Repositorio:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
Idioma:inglés
OAI Identifier:oai:riunet.upv.es:10251/121360
Acceso en línea:https://riunet.upv.es/handle/10251/121360
Access Level:acceso abierto
Palabra clave:Nonautonomous differential equations
Time-averaged differential equations
Quadrature rules
Geometric integration
MATEMATICA APLICADA
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spelling Time-average on the numerical integration of nonautonomous differential equationsBlanes Zamora, Sergio|||0000-0001-5819-8898Nonautonomous differential equationsTime-averaged differential equationsQuadrature rulesGeometric integrationMATEMATICA APLICADA[EN] In this work we show how to numerically integrate nonautonomous differential equations by solving alternate time-averaged differential equations. Given a quadrature rule of order 2s or higher for s = 1, 2, . . . , we show how to build a differential equation with an averaged vector field that is a polynomial function of degree s - 1 in the independent variable, t, and whose solution after one time step agrees with the solution of the original differential equation up to order 2s. Then, any numerical scheme can be used to solve this alternate averaged equation where the vector field is always evaluated at the chosen quadrature rule. We show how to use the Magnus series expansion, adapted to nonlinear problems, to build the formal solution, and this result is valid for any choice of the quadrature rule. This formal solution can be used to build new schemes that must agree with it up to the desired order. For example, we show how to build commutator-free methods from previous results in the literature. All methods can also be written in terms of moment integrals, and each integral can be computed using different quadrature rules. This procedure allows us to build tailored methods for different classes of problems. We illustrate the time-averaged procedure and its efficiency in solving several problems.This work was funded by Ministerio de Economia, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE).Society for Industrial and Applied MathematicsDepartamento de Matemática AplicadaEscuela Técnica Superior de Ingeniería Aeroespacial y Diseño IndustrialInstituto Universitario de Matemática MultidisciplinarMinisterio de Economía y CompetitividadRepositorio Institucional de la Universitat Politècnica de València Riunet20182018-01-01journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://riunet.upv.es/handle/10251/121360reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valénciainstname:Universitat Politècnica de València (UPV)InglésengMinisterio de Economía y Competitividad http://dx.doi.org/10.13039/501100003329 MTM2016-77660-P NUEVOS RETOS EN INTEGRACION NUMERICA: FUNDAMENTOS ALGEBRAICOS, METODOS DE ESCISION, METODOS DE MONTECARLO Y OTRAS APLICACIONESopen accesshttp://purl.org/coar/access_right/c_abf2Reserva de todos los derechoshttp://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:riunet.upv.es:10251/1213602026-06-13T07:49:27Z
dc.title.none.fl_str_mv Time-average on the numerical integration of nonautonomous differential equations
title Time-average on the numerical integration of nonautonomous differential equations
spellingShingle Time-average on the numerical integration of nonautonomous differential equations
Blanes Zamora, Sergio|||0000-0001-5819-8898
Nonautonomous differential equations
Time-averaged differential equations
Quadrature rules
Geometric integration
MATEMATICA APLICADA
title_short Time-average on the numerical integration of nonautonomous differential equations
title_full Time-average on the numerical integration of nonautonomous differential equations
title_fullStr Time-average on the numerical integration of nonautonomous differential equations
title_full_unstemmed Time-average on the numerical integration of nonautonomous differential equations
title_sort Time-average on the numerical integration of nonautonomous differential equations
dc.creator.none.fl_str_mv Blanes Zamora, Sergio|||0000-0001-5819-8898
author Blanes Zamora, Sergio|||0000-0001-5819-8898
author_facet Blanes Zamora, Sergio|||0000-0001-5819-8898
author_role author
dc.contributor.none.fl_str_mv Departamento de Matemática Aplicada
Escuela Técnica Superior de Ingeniería Aeroespacial y Diseño Industrial
Instituto Universitario de Matemática Multidisciplinar
Ministerio de Economía y Competitividad
Repositorio Institucional de la Universitat Politècnica de València Riunet
dc.subject.none.fl_str_mv Nonautonomous differential equations
Time-averaged differential equations
Quadrature rules
Geometric integration
MATEMATICA APLICADA
topic Nonautonomous differential equations
Time-averaged differential equations
Quadrature rules
Geometric integration
MATEMATICA APLICADA
description [EN] In this work we show how to numerically integrate nonautonomous differential equations by solving alternate time-averaged differential equations. Given a quadrature rule of order 2s or higher for s = 1, 2, . . . , we show how to build a differential equation with an averaged vector field that is a polynomial function of degree s - 1 in the independent variable, t, and whose solution after one time step agrees with the solution of the original differential equation up to order 2s. Then, any numerical scheme can be used to solve this alternate averaged equation where the vector field is always evaluated at the chosen quadrature rule. We show how to use the Magnus series expansion, adapted to nonlinear problems, to build the formal solution, and this result is valid for any choice of the quadrature rule. This formal solution can be used to build new schemes that must agree with it up to the desired order. For example, we show how to build commutator-free methods from previous results in the literature. All methods can also be written in terms of moment integrals, and each integral can be computed using different quadrature rules. This procedure allows us to build tailored methods for different classes of problems. We illustrate the time-averaged procedure and its efficiency in solving several problems.
publishDate 2018
dc.date.none.fl_str_mv 2018
2018-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://riunet.upv.es/handle/10251/121360
url https://riunet.upv.es/handle/10251/121360
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Economía y Competitividad http://dx.doi.org/10.13039/501100003329 MTM2016-77660-P NUEVOS RETOS EN INTEGRACION NUMERICA: FUNDAMENTOS ALGEBRAICOS, METODOS DE ESCISION, METODOS DE MONTECARLO Y OTRAS APLICACIONES
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Reserva de todos los derechos
http://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Reserva de todos los derechos
http://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
dc.source.none.fl_str_mv reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
instname:Universitat Politècnica de València (UPV)
instname_str Universitat Politècnica de València (UPV)
reponame_str RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
collection RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,223283

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