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A heterogeneous alternating-direction methodfor a micro-macro dilute polymeric fluid model

Published online by Cambridge University Press:  01 August 2009

David J. Knezevic  and
Endre Süli
David J. Knezevic
Affiliation:
OUCL, University of Oxford, Parks Road, Oxford, OX1 3QD, UK. [email protected]; [email protected]
Endre Süli
Affiliation:
OUCL, University of Oxford, Parks Road, Oxford, OX1 3QD, UK. [email protected]; [email protected]
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Abstract

We examine a heterogeneous alternating-direction method for the approximate solution of the FENE Fokker–Planck equation from polymer fluid dynamics and we use this method to solve a coupled (macro-micro) Navier–Stokes–Fokker–Planck system for dilute polymeric fluids. In this context the Fokker–Planck equation is posed on a high-dimensional domain and is therefore challenging from a computational point of view. The heterogeneous alternating-direction scheme combines a spectral Galerkin method for the Fokker–Planck equation in configuration space with a finite element method in physical space to obtain a scheme for the high-dimensional Fokker–Planck equation. Alternating-direction methods have been considered previously in the literature for this problem (e.g. in the work of Lozinski, Chauvière and collaborators [J. Non-Newtonian Fluid Mech.122 (2004) 201–214; Comput. Fluids33 (2004) 687–696; CRM Proc. Lect. Notes41 (2007) 73–89; Ph.D. Thesis (2003); J. Computat. Phys.189 (2003) 607–625]), but this approach has not previously been subject to rigorous numerical analysis. The numerical methods we develop are fully-practical, and we present a range of numerical results demonstrating their accuracy and efficiency. We also examine an advantageous superconvergence property related to the polymeric extra-stress tensor. The heterogeneous alternating-direction method is well suited to implementation on a parallel computer, and we exploit this fact to make large-scale computations feasible.

Keywords

Multiscale modellingkinetic modelsdilute polymersalternating-direction methodsspectral methodsfinite element methodshigh-performance computing.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 6 , November 2009 , pp. 1117 - 1156
Copyright
© EDP Sciences, SMAI, 2009

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References

Ammar, A., Mokdad, B., Chinesta, F. and Keunings, R., A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J. Non-Newtonian Fluid Mech. 139 (2006) 153–176. CrossRef
Ammar, A., Mokdad, B., Chinesta, F. and Keunings, R., A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part II: Transient simulation using space-time separated representations. J. Non-Newtonian Fluid Mech. 144 (2007) 98–121. CrossRef
S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith and H. Zhang, PETSc users manual. Tech. Rep. ANL-95/11 – Revision 2.1.5, Argonne National Laboratory (2004).
Barrett, J.W. and Süli, E., Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off. Math. Models Methods Appl. Sci. 18 (2008) 935–971. CrossRef
Bialecki, B. and Fernandes, R., An orthogonal spline collocation alternating direction implicit Crank-Nicolson method for linear parabolic problems on rectangles. SIAM J. Numer. Anal. 36 (1999) 1414–1434. CrossRef
Bochev, P.B., Gunzburger, M.D. and Shadid, J.N., Stability of the SUPG finite element method for transient advection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 2301–2323. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Second Edn., Springer (2002).
Celia, M. and Pinder, G., An analysis of alternating-direction methods for parabolic equations. Numer. Methods Part. Differ. Equ. 1 (1985) 57–70. CrossRef
Celia, M. and Pinder, G., Generalized alternating-direction collocation methods for parabolic equations. I. Spatially varying coefficients. Numer. Methods Partial Differ. Equ. 3 (1990) 193–214. CrossRef
Chauvière, C. and Lozinski, A., Simulation of complex viscoelastic flows using Fokker–Planck equation: 3D FENE model. J. Non-Newtonian Fluid Mech. 122 (2004) 201–214. CrossRef
Chauvière, C. and Lozinski, A., Simulation of dilute polymer solutions using a Fokker–Planck equation. Comput. Fluids 33 (2004) 687–696. CrossRef
Clément, P., Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Anal. Numér. 9 (1975) 77–84.
Delaunay, P., Lozinski, A. and Owens, R.G., Sparse tensor-product Fokker–Planck-based methods for nonlinear bead-spring chain models of dilute polymer solutions. CRM Proc. Lect. Notes 41 (2007) 73–89. CrossRef
J. Douglas and T. Dupont, Alternating-direction Galerkin methods on rectangles. Numer. Solution Partial Differ. Equ. II (SYNSPADE 1970) (1971) 133–214.
Eisen, H., Heinrichs, W. and Witsch, K., Spectral collocation methods and polar coordinate singularities. J. Comput. Phys. 96 (1991) 241–257. CrossRef
H. Elman, D. Silvester and A. Wathen, Finite elements and fast iterative solvers. Oxford Science Publications, UK (2005).
Helzel, C. and Otto, F., Multiscale simulations of suspensions of rod-like molecules. J. Comp. Phys. 216 (2006) 52–75. CrossRef
Huang, W. and Guo, B., Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations. Appl. Math. Mech. 29 (2008) 453–476 (English Ed.). CrossRef
Jourdain, B., Lelièvre, T. and Le Bris, C., Existence of solution for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal. 209 (2004) 162–193. CrossRef
Kirk, B.S., Peterson, J.W., Stogner, R.M. and Carey, G.F., libMesh: A C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 23 (2006) 237–254. CrossRef
D.J. Knezevic, Analysis and implementation of numerical methods for simulating dilute polymeric fluids. Ph.D. Thesis, University of Oxford, UK (2008), http://www.comlab.ox.ac.uk/people/David.Knezevic.
Knezevic, D.J. and Süli, E., Spectral Galerkin approximation of Fokker–Planck equations with unbounded drift. ESAIM: M2AN 43 (2009) 445–485. CrossRef
A.N. Kolmogorov, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 (1931).
Li, T. and Zhang, P., Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci. 5 (2007) 1–51. CrossRef
Liu, C. and Liu, H., Boundary conditions for the microscopic FENE models. SIAM J. Appl. Math. 68 (2008) 1304–1315. CrossRef
A. Lozinski, Spectral methods for kinetic theory models of viscoelastic fluids. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, Suisse (2003).
Lozinski, A. and Chauvière, C., A fast solver for Fokker–Planck equation applied to viscoelastic flows calculation: 2D FENE model. J. Computat. Phys. 189 (2003) 607–625. CrossRef
Lyness, J.N. and Jespersen, D., Moderate degree symmetric quadrature rules for the triangle. J. Inst. Math. Appl. 15 (1975) 19–32. CrossRef
Matsushima, T. and Marcus, P.S., A spectral method for polar coordinates. J. Comput. Phys. 120 (1995) 365–374. CrossRef
H.C. Öttinger, Stochastic Processes in Polymeric Fluids. Springer (1996).
R.G. Owens and T.N. Phillips, Computational Rheology. Imperial College Press (2002).
Schwab, C., Süli, E. and Todor, R.A., Sparse finite element approximation of high-dimensional transport-dominated diffusion problems. ESAIM: M2AN 42 (2008) 777–820. CrossRef
Scott, L.R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. CrossRef
Verkley, W.T.M., A spectral model for two-dimensional incompressible fluid flow in a circular basin. I. Mathematical formulation. J. Comput. Phys. 136 (1997) 100–114. CrossRef
N.J. Walkington, Quadrature on simplices of arbitrary dimension. http://www.math.cmu.edu/ nw0z/publications/00-CNA-023/023abs/.
Warner, H.R., Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Ind. Eng. Chem. Fundamentals 11 (1972) 379–387. CrossRef
Zhang, H. and Zhang, P., Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. 181 (2006) 373–400. CrossRef