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

David J. Knezevic; Endre Süli

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 6, page 1117-1156
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Knezevic, David J., and Süli, Endre. "A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model." ESAIM: Mathematical Modelling and Numerical Analysis 43.6 (2009): 1117-1156. <http://eudml.org/doc/250613>.

@article{Knezevic2009,
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. },
author = {Knezevic, David J., Süli, Endre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Multiscale modelling; kinetic models; dilute polymers; alternating-direction methods; spectral methods; finite element methods; high-performance computing.; multiscale modelling; high-performance computing},
language = {eng},
month = {8},
number = {6},
pages = {1117-1156},
publisher = {EDP Sciences},
title = {A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model},
url = {http://eudml.org/doc/250613},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Knezevic, David J.
AU - Süli, Endre
TI - A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/8//
PB - EDP Sciences
VL - 43
IS - 6
SP - 1117
EP - 1156
AB - 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.
LA - eng
KW - Multiscale modelling; kinetic models; dilute polymers; alternating-direction methods; spectral methods; finite element methods; high-performance computing.; multiscale modelling; high-performance computing
UR - http://eudml.org/doc/250613
ER -

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Citations in EuDML Documents

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  1. John W. Barrett, Endre Süli, Finite element approximation of kinetic dilute polymer models with microscopic cut-off
  2. John W. Barrett, Endre Süli, Finite element approximation of kinetic dilute polymer models with microscopic cut-off
  3. John W. Barrett, Endre Süli, Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers
  4. John W. Barrett, Endre Süli, Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

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