Handle URI:
http://hdl.handle.net/10754/600027
Title:
Time-Discrete Higher-Order ALE Formulations: Stability
Authors:
Bonito, Andrea; Kyza, Irene; Nochetto, Ricardo H.
Abstract:
Arbitrary Lagrangian Eulerian (ALE) formulations deal with PDEs on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity. This arbitrary extension has no effect on the stability of the PDE but may influence that of a discrete scheme. We examine this critical issue for higher-order time stepping without space discretization. We propose time-discrete discontinuous Galerkin (dG) numerical schemes of any order for a time-dependent advection-diffusion-model problem in moving domains, and study their stability properties. The analysis hinges on the validity of the Reynold's identity for dG. Exploiting the variational structure and assuming exact integration, we prove that our conservative and nonconservative dG schemes are equivalent and unconditionally stable. The same results remain true for piecewise polynomial ALE maps of any degree and suitable quadrature that guarantees the validity of the Reynold's identity. This approach generalizes the so-called geometric conservation law to higher-order methods. We also prove that simpler Runge-Kutta-Radau methods of any order are conditionally stable, that is, subject to a mild ALE constraint on the time steps. Numerical experiments corroborate and complement our theoretical results. © 2013 Society for Industrial and Applied Mathematics.
Citation:
Bonito A, Kyza I, Nochetto RH (2013) Time-Discrete Higher-Order ALE Formulations: Stability. SIAM J Numer Anal 51: 577–604. Available: http://dx.doi.org/10.1137/120862715.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Numerical Analysis
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
Jan-2013
DOI:
10.1137/120862715
Type:
Article
ISSN:
0036-1429; 1095-7170
Sponsors:
This author was partially supported by NSF grant DMS-0914977 and by Award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).This author was partially supported by NSF grants DMS-0807811 and DMS-0807815.This author was partially supported by NSF grants DMS-0807811 and DMS-1109325.
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Full metadata record

DC FieldValue Language
dc.contributor.authorBonito, Andreaen
dc.contributor.authorKyza, Ireneen
dc.contributor.authorNochetto, Ricardo H.en
dc.date.accessioned2016-02-28T06:34:36Zen
dc.date.available2016-02-28T06:34:36Zen
dc.date.issued2013-01en
dc.identifier.citationBonito A, Kyza I, Nochetto RH (2013) Time-Discrete Higher-Order ALE Formulations: Stability. SIAM J Numer Anal 51: 577–604. Available: http://dx.doi.org/10.1137/120862715.en
dc.identifier.issn0036-1429en
dc.identifier.issn1095-7170en
dc.identifier.doi10.1137/120862715en
dc.identifier.urihttp://hdl.handle.net/10754/600027en
dc.description.abstractArbitrary Lagrangian Eulerian (ALE) formulations deal with PDEs on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity. This arbitrary extension has no effect on the stability of the PDE but may influence that of a discrete scheme. We examine this critical issue for higher-order time stepping without space discretization. We propose time-discrete discontinuous Galerkin (dG) numerical schemes of any order for a time-dependent advection-diffusion-model problem in moving domains, and study their stability properties. The analysis hinges on the validity of the Reynold's identity for dG. Exploiting the variational structure and assuming exact integration, we prove that our conservative and nonconservative dG schemes are equivalent and unconditionally stable. The same results remain true for piecewise polynomial ALE maps of any degree and suitable quadrature that guarantees the validity of the Reynold's identity. This approach generalizes the so-called geometric conservation law to higher-order methods. We also prove that simpler Runge-Kutta-Radau methods of any order are conditionally stable, that is, subject to a mild ALE constraint on the time steps. Numerical experiments corroborate and complement our theoretical results. © 2013 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipThis author was partially supported by NSF grant DMS-0914977 and by Award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).This author was partially supported by NSF grants DMS-0807811 and DMS-0807815.This author was partially supported by NSF grants DMS-0807811 and DMS-1109325.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectALE formulationsen
dc.subjectDG methods in timeen
dc.subjectDiscrete Reynold's identitiesen
dc.subjectDomain velocityen
dc.subjectGeometric conservation lawen
dc.subjectMaterial derivativeen
dc.subjectMoving domainsen
dc.subjectStabilityen
dc.titleTime-Discrete Higher-Order ALE Formulations: Stabilityen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Numerical Analysisen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionFoundation for Research and Technology-Hellas, Heraklion, Greeceen
dc.contributor.institutionUniversity of Maryland, College Park, United Statesen
kaust.grant.numberKUS-C1-016-04en
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