Free-energy-dissipative schemes for the Oldroyd-B model

Abstract

Abstract.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. NonNewtonian 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. Mathematics Subject Classification. 65M12, 76M10, 35B45, 76A10, 35B35.

Existence and Approximation of a (Regularized) Oldroyd-B Model

“…First, we extend previous results in Boyaval et al 7 for a finite element approximation of (P) using essentially the backward Euler scheme in time and based on approximating the pressure and the symmetric conformation tensor by piecewise constants; and the velocity field with continuous piecewise quadratics or a reduced version, where the tangential component on each simplicial edge (d = 2) or face (d = 3) is linear. We show that solutions of this numerical scheme satisfy a discrete free energy bound, which involves the logarithm of the conformation tensor, without any constraint on the time step, whereas a time constraint based on the initial data was required in Boyaval et al 7 in order to ensure that the approximation to the conformation tensor σ remained positive definite. See also Lee and Xu,18 where the difficulties of maintaining the positive definiteness of approximations to σ are also discussed.…”

“…Once again we refer to p267 in Ern and Guermond 9 for the consistency of our stated approximation of the stress convection term, see also Boyaval et al 7 .…”

“…poor numerical scheme or the lack of existence of a solution to (P) itself. In addition in Boyaval et al, 7 finite element approximations of (P) such as (P Δt h ), approximating the primitive variables (u, p, σ), are compared with finite element approximations of the log-formulation of (P), introduced in Fattal and Kupferman, 10 which is based on the variables (u, p, ψ), where ψ = ln σ. The equivalent free energy estimate for this log-formulation is based on testing the Navier-Stokes equation with u as before, but the log-form of the stress equation with (exp ψ − I).…”

“…Whereas the free energy estimate for (P) requires σ to be positive definite, due to the testing with ln σ, the free energy estimate for the log-formulation requires no such constraint. In Boyaval et al 7 a constraint, based on the initial data, was required on the time step in order to ensure that the approximation to σ remained positive definite for schemes such as (P Δt h ) approximating (P); whereas existence of a solution to finite element approximations of the log-formulation, and satisfying a discrete log-form of the free energy estimate, were shown for any choice of time step. It was suggested in Boyaval et al…”

“…We note that a piecewise constant approximation of the conformation tensor was necessary in Boyaval et al 7 in order to treat the advection term in (1.1c) and still obtain a discrete free energy bound.…”