  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, which have been reported to be numerically more stable
than discretizations of the usual formulation in some benchmark problems. Our
analysis gives some tracks to understand these numerical observations.
