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Issue ESAIM: M2AN
Volume 43, Number 3, May-June 2009
Page(s) 523 - 561
DOI 10.1051/m2an/2009008
Published online 08 April 2009

ESAIM: M2AN 43 (2009) 523-561
DOI: 10.1051/m2an/2009008

Free-energy-dissipative schemes for the Oldroyd-B model

Sébastien Boyaval1, 2, Tony Lelièvre1, 2 and Claude Mangoubi1, 2, 3

1  CERMICS, École Nationale des Ponts et Chaussées (ParisTech/Université Paris-Est), 6 & 8 avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée Cedex 2, France. boyaval@cermics.enpc.fr; lelievre@cermics.enpc.fr; mangoubi@cermics.enpc.fr
2  MICMAC team-project, INRIA, Domaine de Voluceau, BP. 105, Rocquencourt, 78153 Le Chesnay Cedex, France.
3  Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel.

Received January 15, 2008. Revised September 16, 2008. Published online April 8, 2009.

Abstract
In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman in [J. Non-Newtonian Fluid Mech. 123 (2004) 281–285], for which solutions in some benchmark problems have been obtained beyond the limiting Weissenberg numbers for the standard scheme (see [Hulsen et al. J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]). Our analysis gives some tracks to understand these numerical observations.


Mathematics Subject Classification. 65M12, 76M10, 35B45, 76A10, 35B35

Key words: Viscoelastic fluids, Weissenberg number, stability, entropy, finite elements methods, discontinuous Galerkin method, characteristic method.


© EDP Sciences, SMAI 2009


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