An ADER-type scheme for a class of equations arising from the water-wave theory

In this work we propose a numerical strategy to solve a family of partial differential equations arising from the water-wave theory. These problems may contain four terms; a source which is an algebraic function of the solution, a convective part involving first order spatial derivatives of the solu...

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Autores: Montecinos, G.I., López-Rios, J.C., Lecaros, R., Ortega, J.H., Toro, E.F.
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2016
País:España
Institución:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/257
Acceso en línea:http://hdl.handle.net/20.500.11824/257
Access Level:acceso abierto
Palabra clave:ADER schemes
Finite volume schemes
Generalized Riemann problems
Water waves equations
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spelling An ADER-type scheme for a class of equations arising from the water-wave theoryMontecinos, G.I.López-Rios, J.C.Lecaros, R.Ortega, J.H.Toro, E.F.ADER schemesFinite volume schemesGeneralized Riemann problemsWater waves equationsIn this work we propose a numerical strategy to solve a family of partial differential equations arising from the water-wave theory. These problems may contain four terms; a source which is an algebraic function of the solution, a convective part involving first order spatial derivatives of the solution, a diffusive part involving second order spatial derivatives and the transient part. Unlike partial differential equations of hyperbolic or parabolic type, where the transient part is the time derivative of the solution, here the transient part can contain mixed time and space derivatives.In [Zambra et al. International Journal for Numerical Methods in Engineering 89(2):227-240, 2012], the authors proposed a globally implicit strategy to solve the Richards equation. In that case, transient terms consisted of algebraic expressions of the solution. Motivated by this work, we propose a one-step finite volume method to deal with problems in which transient terms are differential operators. Here, a locally implicit formulation is investigated, which is based on the ADER philosophy. The scheme is divided in three steps: i) a polynomial reconstruction of the data; ii) solutions to Generalized Riemann Problems (GRP); iii) the solution of differential problems. Note that steps i) and ii), are those of conventional ADER schemes for conservation laws. Advantages of the present approach include the possibility to construct high-order approximations in both space and time, for which existing methodologies for hyperbolic problems can be applied. The differential problems associated to the transient term can be non-linear and numerical strategies can be adopted to deal wit it. Convergence of the scheme is proved rigorously and an empirical convergence rates assessment is carried out in order to illustrate the high space and time accuracy of the present scheme.201620162016info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/257reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttp://www.sciencedirect.com/science/article/pii/S0045793016301128Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/2572026-06-19T12:47:47Z
dc.title.none.fl_str_mv An ADER-type scheme for a class of equations arising from the water-wave theory
title An ADER-type scheme for a class of equations arising from the water-wave theory
spellingShingle An ADER-type scheme for a class of equations arising from the water-wave theory
Montecinos, G.I.
ADER schemes
Finite volume schemes
Generalized Riemann problems
Water waves equations
title_short An ADER-type scheme for a class of equations arising from the water-wave theory
title_full An ADER-type scheme for a class of equations arising from the water-wave theory
title_fullStr An ADER-type scheme for a class of equations arising from the water-wave theory
title_full_unstemmed An ADER-type scheme for a class of equations arising from the water-wave theory
title_sort An ADER-type scheme for a class of equations arising from the water-wave theory
dc.creator.none.fl_str_mv Montecinos, G.I.
López-Rios, J.C.
Lecaros, R.
Ortega, J.H.
Toro, E.F.
author Montecinos, G.I.
author_facet Montecinos, G.I.
López-Rios, J.C.
Lecaros, R.
Ortega, J.H.
Toro, E.F.
author_role author
author2 López-Rios, J.C.
Lecaros, R.
Ortega, J.H.
Toro, E.F.
author2_role author
author
author
author
dc.subject.none.fl_str_mv ADER schemes
Finite volume schemes
Generalized Riemann problems
Water waves equations
topic ADER schemes
Finite volume schemes
Generalized Riemann problems
Water waves equations
description In this work we propose a numerical strategy to solve a family of partial differential equations arising from the water-wave theory. These problems may contain four terms; a source which is an algebraic function of the solution, a convective part involving first order spatial derivatives of the solution, a diffusive part involving second order spatial derivatives and the transient part. Unlike partial differential equations of hyperbolic or parabolic type, where the transient part is the time derivative of the solution, here the transient part can contain mixed time and space derivatives.In [Zambra et al. International Journal for Numerical Methods in Engineering 89(2):227-240, 2012], the authors proposed a globally implicit strategy to solve the Richards equation. In that case, transient terms consisted of algebraic expressions of the solution. Motivated by this work, we propose a one-step finite volume method to deal with problems in which transient terms are differential operators. Here, a locally implicit formulation is investigated, which is based on the ADER philosophy. The scheme is divided in three steps: i) a polynomial reconstruction of the data; ii) solutions to Generalized Riemann Problems (GRP); iii) the solution of differential problems. Note that steps i) and ii), are those of conventional ADER schemes for conservation laws. Advantages of the present approach include the possibility to construct high-order approximations in both space and time, for which existing methodologies for hyperbolic problems can be applied. The differential problems associated to the transient term can be non-linear and numerical strategies can be adopted to deal wit it. Convergence of the scheme is proved rigorously and an empirical convergence rates assessment is carried out in order to illustrate the high space and time accuracy of the present scheme.
publishDate 2016
dc.date.none.fl_str_mv 2016
2016
2016
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info:eu-repo/semantics/acceptedVersion
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dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/257
url http://hdl.handle.net/20.500.11824/257
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv http://www.sciencedirect.com/science/article/pii/S0045793016301128
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv openAccess
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dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
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