Positivity-preserving methods for ordinary differential equations

[EN] Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main familie...

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Autores: Blanes Zamora, Sergio|||0000-0001-5819-8898, Iserles, Arieh, MacNamara, Shev
Tipo de recurso: artículo
Fecha de publicación:2022
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:10251/196131
Acceso en línea:https://riunet.upv.es/handle/10251/196131
Access Level:acceso abierto
Palabra clave:Positivity-preserving methods
Graph Laplacian matrices
Exponential integrators
Magnus integrators
MATEMATICA APLICADA
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spelling Positivity-preserving methods for ordinary differential equationsBlanes Zamora, Sergio|||0000-0001-5819-8898Iserles, AriehMacNamara, ShevPositivity-preserving methodsGraph Laplacian matricesExponential integratorsMagnus integratorsMATEMATICA APLICADA[EN] Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.The authors thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Geometry, compatibility and structure preservation in computational differential equations" when work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1. S.B. has been supported by project PID2019-104927GB-C21 (AEI/FEDER, UE).EDP SciencesDepartamento de Matemática AplicadaEscuela Técnica Superior de Ingeniería Aeroespacial y Diseño IndustrialInstituto Universitario de Matemática MultidisciplinarEuropean Regional Development FundMinisterio de Ciencia e InnovaciónEngineering and Physical Sciences Research Council, Reino UnidoRepositorio Institucional de la Universitat Politècnica de València Riunet20222022-08-12journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://riunet.upv.es/handle/10251/196131reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valénciainstname:Universitat Politècnica de València (UPV)InglésengAgencia Estatal de Investigación http://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PID2019-104927GB-C21 METODOS DE INTEGRACION GEOMETRICA PARA PROBLEMAS CUANTICOS, MECANICA CELESTE Y SIMULACIONES MONTECARLO IEngineering and Physical Sciences Research Council, Reino Unido https://doi.org/10.13039/501100000266 EP%2FR014604%2F1Ministerio de Ciencia e Innovación http://dx.doi.org/10.13039/501100004837 PID2019-104927GB-C21 METODOS DE INTEGRACION GEOMETRICA PARA PROBLEMAS CUANTICOS, MECANICA CELESTE Y SIMULACIONES MONTECARLO Iopen accesshttp://purl.org/coar/access_right/c_abf2Reconocimiento (by)http://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:riunet.upv.es:10251/1961312026-06-13T07:49:27Z
dc.title.none.fl_str_mv Positivity-preserving methods for ordinary differential equations
title Positivity-preserving methods for ordinary differential equations
spellingShingle Positivity-preserving methods for ordinary differential equations
Blanes Zamora, Sergio|||0000-0001-5819-8898
Positivity-preserving methods
Graph Laplacian matrices
Exponential integrators
Magnus integrators
MATEMATICA APLICADA
title_short Positivity-preserving methods for ordinary differential equations
title_full Positivity-preserving methods for ordinary differential equations
title_fullStr Positivity-preserving methods for ordinary differential equations
title_full_unstemmed Positivity-preserving methods for ordinary differential equations
title_sort Positivity-preserving methods for ordinary differential equations
dc.creator.none.fl_str_mv Blanes Zamora, Sergio|||0000-0001-5819-8898
Iserles, Arieh
MacNamara, Shev
author Blanes Zamora, Sergio|||0000-0001-5819-8898
author_facet Blanes Zamora, Sergio|||0000-0001-5819-8898
Iserles, Arieh
MacNamara, Shev
author_role author
author2 Iserles, Arieh
MacNamara, Shev
author2_role author
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
European Regional Development Fund
Ministerio de Ciencia e Innovación
Engineering and Physical Sciences Research Council, Reino Unido
Repositorio Institucional de la Universitat Politècnica de València Riunet
dc.subject.none.fl_str_mv Positivity-preserving methods
Graph Laplacian matrices
Exponential integrators
Magnus integrators
MATEMATICA APLICADA
topic Positivity-preserving methods
Graph Laplacian matrices
Exponential integrators
Magnus integrators
MATEMATICA APLICADA
description [EN] Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.
publishDate 2022
dc.date.none.fl_str_mv 2022
2022-08-12
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/10251/196131
url https://riunet.upv.es/handle/10251/196131
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación http://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PID2019-104927GB-C21 METODOS DE INTEGRACION GEOMETRICA PARA PROBLEMAS CUANTICOS, MECANICA CELESTE Y SIMULACIONES MONTECARLO I
Engineering and Physical Sciences Research Council, Reino Unido https://doi.org/10.13039/501100000266 EP%2FR014604%2F1
Ministerio de Ciencia e Innovación http://dx.doi.org/10.13039/501100004837 PID2019-104927GB-C21 METODOS DE INTEGRACION GEOMETRICA PARA PROBLEMAS CUANTICOS, MECANICA CELESTE Y SIMULACIONES MONTECARLO I
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Reconocimiento (by)
http://creativecommons.org/licenses/by/4.0/
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
Reconocimiento (by)
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv EDP Sciences
publisher.none.fl_str_mv EDP Sciences
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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