Issue ESAIM: M2AN Volume 43, Number 6, November-December 2009 Page(s) 1117 - 1156 DOI 10.1051/m2an/2009034 Published online 01 August 2009

ESAIM: M2AN 43 (2009) 1117-1156
DOI: 10.1051/m2an/2009034

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

David J. Knezevic and Endre Süli

OUCL, University of Oxford, Parks Road, Oxford, OX1 3QD, UK. davek@comlab.ox.ac.uk; endre.suli@comlab.ox.ac.uk

Received October 23, 2008. Revised March 17, 2009. Published online August 1st, 2009.

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. Fluids 33 (2004) 687–696; CRM Proc. Lect. Notes 41 (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.

Mathematics Subject Classification. 65M70, 65M12, 35K20, 82C31, 82D60.

Key words: Multiscale modelling, kinetic models, dilute polymers, alternating-direction methods, spectral methods, finite element methods, high-performance computing.


© EDP Sciences, SMAI 2009

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