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.

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 }, pages = {523--561}, publisher = {EDP-Sciences}, volume = {43}, number = {3}, year = {2009}, doi = {10.1051/m2an/2009008}, zbl = {1167.76018}, mrnumber = {2536248}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2009008/} }

TY - JOUR AU - Boyaval, Sébastien AU - Lelièvre, Tony AU - Mangoubi, Claude TI - Free-energy-dissipative schemes for the Oldroyd-B model JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2009 SP - 523 EP - 561 VL - 43 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2009008/ DO - 10.1051/m2an/2009008 LA - en ID - M2AN_2009__43_3_523_0 ER -

%0 Journal Article %A Boyaval, Sébastien %A Lelièvre, Tony %A Mangoubi, Claude %T Free-energy-dissipative schemes for the Oldroyd-B model %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2009 %P 523-561 %V 43 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2009008/ %R 10.1051/m2an/2009008 %G en %F 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 , Volume 43 (2009) no. 3, pp. 523-561. doi : 10.1051/m2an/2009008. http://www.numdam.org/articles/10.1051/m2an/2009008/

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