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

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: 15 January 2008
Revised: 16 September 2008

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.