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.