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.
Mots clés : 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},
publisher = {EDP-Sciences},
volume = {43},
number = {3},
year = {2009},
doi = {10.1051/m2an/2009008},
mrnumber = {2536248},
zbl = {1167.76018},
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: Modélisation mathématique et analyse numérique 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: Modélisation mathématique et analyse numérique %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: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 523-561. doi : 10.1051/m2an/2009008. http://www.numdam.org/articles/10.1051/m2an/2009008/
[1] and , Quadratic velocity/linear pressure Stokes elements, in Advances in Computer Methods for Partial Differential Equations, Volume VII, R. Vichnevetsky and R.S. Steplemen Eds. (1992).
[2] , and , Coil-stretch transition and the breakdown of computations for viscoelastic fluid flow around a confined cylinder. J. Rheol. 52 (2008) 197-223.
[3] and , Existence of approximate solutions and error bounds for viscoelastic fluid flow: Characteristics method. Comput. Methods Appl. Mech. Engrg. 148 (1997) 39-52. | MR | Zbl
[4] and , Convergence of a finite element approximation to a regularized Oldroyd-B model (in preparation).
[5] , and , Existence of global weak solutions for some polymeric flow models. Math. Mod. Meth. Appl. Sci. 15 (2005) 939-983. | MR | Zbl
[6] and , Thermodynamics of flowing systems with internal microstructure. Oxford University Press (1994). | MR
[7] , and , GLS and EVSS methods for a three fields Stokes problem arising from viscoelastic flows. Comp. Meth. Appl. Mech. Eng. 190 (2001) 3893-3914. | MR | Zbl
[8] and , On the stabilization of finite element approximations of the Stokes equations, in Efficient Solution of Elliptic System, W. Hackbusch Ed. (1984) 11-19. | MR | Zbl
[9] , and , Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217-235. | MR | Zbl
[10] , and , Recent results on mixed finite element methods for second order elliptic problems, in Vistas in Applied Mathematics: Numerical Analysis, Atmospheric Sciences, Immunology, A.V. Balakrishnan, A.A. Dorodnitsyn and J.L. Lions Eds. (1986) 25-43. | MR | Zbl
[11] , Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comp. Meth. Appl. Mech. Engrg. 156 (1998) 185-210. | MR | Zbl
[12] and , Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 3 (1973) 33-75. | Numdam | MR | Zbl
[13] and , Theory and practice of finite elements. Springer Verlag, New-York (2004). | MR | Zbl
[14] and , Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123 (2004) 281-285. | Zbl
[15] and , Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. J. Non-Newtonian Fluid Mech. 126 (2005) 23-37. | Zbl
[16] , , and , Global stability of equilibrium manifolds, and “peaking'' behavior in quadratic differential systems related to viscoelastic models. J. Non-Newtonian Fluid Mech. 144 (2007) 30-41.
[17] , and , Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, in Handbook of Numerical Analysis, Vol. 8, P.G. Ciarlet et al. Eds., Elsevier (2002) 543-661. | MR | Zbl
[18] , Stabilization of Galerkin approximations of transport equations by subgrid modeling. ESAIM: M2AN 33 (1999) 1293-1316. | Numdam | MR | Zbl
[19] and , Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlin. Anal. TMA 15 (1990) 849-869. | MR | Zbl
[20] and , New entropy estimates for the Oldroyd-B model, and related models. Commun. Math. Sci. 5 (2007) 906-916. | Zbl
[21] and , A new finite element formulation for CFD: VII the Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces. Comp. Meth. App. Mech. Eng. 65 (1987) 85-96. | MR | Zbl
[22] , A sufficient condition for a positive definite configuration tensor in differential models. J. Non-Newtonian Fluid Mech. 38 (1990) 93-100.
[23] , and , Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms. J. Non-Newtonian Fluid Mech. 127 (2005) 27-39.
[24] , , and , Long-time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal. 181 (2006) 97-148. | MR | Zbl
[25] and , Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58 (1992) 1-10. | MR | Zbl
[26] , Numerical instability of time-dependent flows. J. Non-Newtonian Fluid Mech. 43 (1992) 229-246. | Zbl
[27] , Simulation of viscoelastic fluid flow, in Fundamentals of Computer Modeling for Polymer Processing, C. Tucker Ed., Hanse (1989) 402-470.
[28] , A survey of computational rheology, in Proc. 13th Int. Congr. on Rheology, D.M. Binding et al Eds., British Society of Rheology (2000) 7-14.
[29] , and , A Beale-Kato-Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime. Comm. Math. Sci. 6 (2008) 235-256. | MR | Zbl
[30] , Finite element analysis of planar 4:1 contraction flow with the tensor-logarithmic formulation of differential constitutive equations. Korea-Australia Rheology Journal 16 (2004) 183-191.
[31] and , Stability constraints in the formulation of viscoelastic constitutive equations. J. Non-Newtonian Fluid Mech. 58 (1995) 25-46.
[32] and , New formulations positivity preserving discretizations and stability analysis for non-Newtonian flow models. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1180-1206. | MR | Zbl
[33] , Analysis of simple constitutive equations for viscoelastic liquids. J. Non-Newton. Fluid Mech. 42 (1992) 323-350. | Zbl
[34] , and , On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math. 58 (2005) 1437-1471. | MR | Zbl
[35] and , Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math., Ser. B 21 (2000) 131-146. | MR | Zbl
[36] and , An energy estimate for the Oldroyd-B model: theory and applications. J. Non-Newtonian Fluid Mech. 112 (2003) 161-176. | Zbl
[37] and , Introduction à la théorie des groupes de Lie classiques. Hermann (1986). | MR | Zbl
[38] , and , Convergence analysis of the Lagrange-Galerkin method with non-exact integration. RAIRO Modél. Math. Anal. Numér. 22 (1988) 625-653. | Numdam | MR | Zbl
[39] , Beyond Equilibrium Thermodynamics. Wiley (2005).
[40] , On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math. 3 (1982) 309-332. | MR | Zbl
[41] and , Do we understand the physics in the constitutive equation? J. Non-Newtonian Fluid Mech. 29 (1988) 37-55.
[42] , Non integrable extra stress tensor solution for a flow in a bounded domain of an Oldroyd fluid. Acta Mech. 135 (1999) 95-99. | MR | Zbl
[43] and , Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19 (1985) 111-143. | Numdam | MR | Zbl
[44] , Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math. 53 (1988) 459-483. | MR | Zbl
[45] , Sur l'approximation des équations de Navier-Stokes. C. R. Acad. Sci. Paris, Sér. A 262 (1966) 219-221. | MR | Zbl
[46] and , Emergence of singular structures in Oldroyd-B fluids. Phys. Fluids 19 (2007) 103103.
[47] and , Thermodynamics of viscoelastic fluids: the temperature equation. J. Rheol. 42 (1998) 999-1019.
[48] , and , The backward-tracking lagrangian particle method for transient viscoelastic flows. J. Non-Newtonian Fluid Mech. 91 (2000) 273-295. | Zbl
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