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.

Classification: 65M12, 76M10, 35B45, 76A10, 35B35

Keywords: viscoelastic fluids, Weissenberg number, stability, entropy, finite elements methods, discontinuous Galerkin method, characteristic method

@article{M2AN_2009__43_3_523_0, author = {Boyaval, S\'ebastien and Leli\`evre, Tony and Mangoubi, Claude}, title = {Free-energy-dissipative schemes for the Oldroyd-B model}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {43}, number = {3}, year = {2009}, pages = {523-561}, doi = {10.1051/m2an/2009008}, zbl = {1167.76018}, mrnumber = {2536248}, language = {en}, url = {http://www.numdam.org/item/M2AN_2009__43_3_523_0} }

Boyaval, Sébastien; Lelièvre, Tony; Mangoubi, Claude. Free-energy-dissipative schemes for the Oldroyd-B model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 3, pp. 523-561. doi : 10.1051/m2an/2009008. http://www.numdam.org/item/M2AN_2009__43_3_523_0/

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