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Efficient Stabilization of Advection Terms Involved in Separated Representations of Boltzmann and Fokker-Planck Equations

Published online by Cambridge University Press:  30 April 2015

F. Chinesta ,
E. Abisset-Chavanne ,
A. Ammar  and
E. Cueto
F. Chinesta*
Affiliation:
GeM Institute, Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France
E. Abisset-Chavanne
Affiliation:
GeM Institute, Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France
A. Ammar
Affiliation:
Arts et Métiers ParisTech, 2 Boulevard du Ronceray, BP 93525, F-49035 Angers cedex 01, France
E. Cueto
Affiliation:
Aragon Institute of Engineering Research, Universidad de Zaragoza, Maria de Luna, 7, E-50018 Zaragoza, Spain
*
*Corresponding author. Email addresses: Francisco.Chinesta@ec-nantes.fr (F. Chinesta), Emmanuelle.Abisset-chavanne@ec-nantes.fr (E. Abisset-Chavanne), Amine.AMMAR@ensam.eu (A. Ammar), ecueto@unizar.es (E. Cueto)
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Abstract

The fine description of complex fluids can be carried out by describing the evolution of each individual constituent (e.g. each particle, each macromolecule, etc.). This procedure, despite its conceptual simplicity, involves many numerical issues, the most challenging one being that related to the computing time required to update the system configuration by describing all the interactions between the different individuals. Coarse grained approaches allow alleviating the just referred issue: the system is described by a distribution function providing the fraction of entities that at certain time and position have a particular conformation. Thus, mesoscale models involve many different coordinates, standard space and time, and different conformational coordinates whose number and nature depend on the particular system considered. Balance equation describing the evolution of such distribution function consists of an advection-diffusion partial differential equation defined in a high dimensional space. Standard mesh-based discretization techniques fail at solving high-dimensional models because of the curse of dimensionality. Recently the authors proposed an alternative route based on the use of separated representations. However, until now these approaches were unable to address the case of advection dominated models due to stabilization issues. In this paper this issue is revisited and efficient procedures for stabilizing the advection operators involved in the Boltzmann and Fokker-Planck equation within the PGD framework are proposed.

Keywords

35G1578M1578M2578M3435Q8435Q20Boltzmann equationFokker-Planck equationproper generalized decompositionmultidimensional modelsconvective stabilization
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 4 , April 2015 , pp. 975 - 1006
Copyright
Copyright © Global-Science Press 2015 

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