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A heterogeneous alternating-direction method for 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. davek@comlab.ox.ac.uk; endre.suli@comlab.ox.ac.uk
Endre Süli
Affiliation:
OUCL, University of Oxford, Parks Road, Oxford, OX1 3QD, UK. davek@comlab.ox.ac.uk; endre.suli@comlab.ox.ac.uk
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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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