Commun. Comput. Phys.,
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Volume 4.


Colocated Finite Volume Schemes for Fluid Flows

S. Faure 1, J. Laminie 2, R. Temam 3*

1 Laboratoire de Mathematiques, Universite Paris-Sud, Batiment 425, 91405 Orsay, France.
2 Laboratoire de Mathematiques, Universite Paris-Sud, Batiment 425, 91405 Orsay, France; and GRIMAAG Guadeloupe, Universite des Antilles et de la Guyane, Campus de Fouillole, BP 592, 97157 Pointe a Pitre Cedex, France.
3 Laboratoire de Mathematiques, Universite Paris-Sud, Batiment 425, 91405 Orsay, France; and Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA.

Received 11 September 2007; Accepted (in revised version) 10 February 2008
Available online 27 February 2008

Abstract

Our aim in this article is to improve the understanding of the colocated finite volume schemes for the incompressible Navier-Stokes equations. When all the variables are colocated, that means here when the velocities and the pressure are computed at the same place (at the centers of the control volumes), these unknowns must be properly coupled. Consequently, the choice of the time discretization and the method used to interpolate the fluxes at the edges of the control volumes are essentials. In the first and second parts of this article, two different time discretization schemes are considered with a colocated space discretization and we explain how the unknowns can be correctly coupled. Numerical simulations are presented in the last part of the article.

This paper is not a comparison between staggered grid schemes and colocated schemes (for this, see, e.g., \cite{PERIC1,W01}). We plan, in the future, to use a colocated space discretization and the multilevel method of \cite{FT3} initially applied to the two dimensional Burgers problem, in order to solve the incompressible Navier-Stokes equations. One advantage of colocated schemes is that all variables share the same location, hence, the possibility to use hierarchical space discretizations more easily when multilevel methods are used. For this reason, we think that it is important to study this family of schemes.

AMS subject classifications: 76M12, 76D05, 68U120, 74S10, 74H15

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Key words: Finite volumes, colocated scheme.

*Corresponding author.
Email: sylvain.faure@math.u-psud.fr (S. Faure), jacques.laminie@ univ-ag.fr (J. Laminie), temam@indiana.edu (R. Temam)
 

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