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| HAL : hal-00380659, version 1 |
| DOI : 10.1137/090757940 |
| Fiche détaillée |  Récupérer au format |
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| SIAM Journal on Numerical Analysis 48, 6 (2010) 2019-2042 |
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| Explicit Runge--Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems |
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| Erik Burman 1Alexandre Ern 2 |
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| (2010) |
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| We analyze explicit Runge--Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs-type. For the time discretization, we consider explicit second- and third-order Runge--Kutta schemes. We identify a general set of properties on the spatial stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then, we establish $L^2$-norm error estimates with (quasi-)optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for third-order Runge--Kutta schemes and any polynomial degree in space and for second-order Runge--Kutta schemes and first-order polynomials in space. For second-order Runge--Kutta schemes and higher polynomial degrees in space, a tightened 4/3-CFL condition is required. Numerical results are presented for the advection and wave equations. |
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| 1 : | Department of Mathematics |
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University of Sussex |
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| 2 : | Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS) |
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Ecole des Ponts ParisTech |
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| 3 : | REO (INRIA Rocquencourt) |
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INRIA – Laboratoire Jacques-Louis Lions |
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| Domaine | : | Mathématiques/Analyse numérique |
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| First-order PDEs – transient problems – stabilized finite elements – explicit Runge--Kutta schemes – stability – convergence |
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| Liste des fichiers attachés à ce document : | |||||
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| hal-00380659, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00380659 | |
| oai:hal.archives-ouvertes.fr:hal-00380659 | |
| Contributeur : Alexandre Ern | |
| Soumis le : Lundi 4 Mai 2009, 12:21:45 | |
| Dernière modification le : Vendredi 4 Novembre 2011, 12:01:45 | |
