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: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {523--561},
year = {2009},
publisher = {EDP Sciences},
volume = {43},
number = {3},
doi = {10.1051/m2an/2009008},
mrnumber = {2536248},
zbl = {1167.76018},
language = {en},
url = {https://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: Modélisation mathématique et analyse numérique PY - 2009 SP - 523 EP - 561 VL - 43 IS - 3 PB - EDP Sciences UR - https://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: Modélisation mathématique et analyse numérique %D 2009 %P 523-561 %V 43 %N 3 %I EDP Sciences %U https://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: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 523-561. doi: 10.1051/m2an/2009008
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