Free-energy-dissipative schemes for the Oldroyd-B model
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 3, pp. 523-561.

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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.

DOI : 10.1051/m2an/2009008
Classification : 65M12, 76M10, 35B45, 76A10, 35B35
Mots-clés : viscoelastic fluids, Weissenberg number, stability, entropy, finite elements methods, discontinuous Galerkin method, characteristic method 
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     title = {Free-energy-dissipative schemes for the {Oldroyd-B} model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {523--561},
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Boyaval, Sébastien; Lelièvre, Tony; Mangoubi, Claude. Free-energy-dissipative schemes for the Oldroyd-B model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 3, pp. 523-561. doi : 10.1051/m2an/2009008. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an/2009008/

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