Comptes Rendus
no. 3
Numerical Analysis
A simultaneous directions parallel algorithm for the Navier–Stokes equations
[Un algorithme parallèlle de directions simultanées pour les équations de Navier–Stokes.]

Présenté par : Roland Glowinski

José R. Galo1 ; Isidoro I. Albarreal2 ; M. Carmen Calzada1 ; José Luis Cruz1 ; Enrique Fernández-Cara2 ; Mercedes Marín1
1 Dpto. Informática y Análisis Numérico, Universidad de Córdoba, Campus de Rabanales, Ed. C2-3, 14071 Córdoba, Spain
2 Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 235-240.

On utilise un algorithme parallèlle en espace et en temps de type directions simultanées pour résoudre les équations de Navier–Stokes. On adapte quelques idées de Lu, Neittaanmäki et Tai, ce qui conduit à une (grande) famille de systèmes différentiels ordinaires linéaires du second ordre qui son indépendants. On présente les résultats de quelques expériences numériques.

We use a parallel algorithm (in time and space) to solve the incompressible Navier–Stokes problem. Adapting some previous ideas by Lu, Neittaanmäki and Tai, the task is reduced to solving a (large) family of independent second-order one-dimensional linear systems. We also present some numerical experiments.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2004.05.009
José R. Galo 1 ; Isidoro I. Albarreal 2 ; M. Carmen Calzada 1 ; José Luis Cruz 1 ; Enrique Fernández-Cara 2 ; Mercedes Marín 1

1 Dpto. Informática y Análisis Numérico, Universidad de Córdoba, Campus de Rabanales, Ed. C2-3, 14071 Córdoba, Spain
2 Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain 
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     title = {A simultaneous directions parallel algorithm for the {Navier{\textendash}Stokes} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {235--240},
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José R. Galo; Isidoro I. Albarreal; M. Carmen Calzada; José Luis Cruz; Enrique Fernández-Cara; Mercedes Marín. A simultaneous directions parallel algorithm for the Navier–Stokes equations. Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 235-240. doi : 10.1016/j.crma.2004.05.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.009/

[1] I. Albarreal; M.C. Calzada; J.L. Cruz; E. Fernández-Cara; J.R. Galo; M. Marín Convergence analysis and error estimates for a parallel algorithm for solving the Navier–Stokes equations, Numer. Math., Volume 93 (2002), pp. 201-221

[2] J.L. Cruz; M.C. Calzada; M. Marín; E. Fernández-Cara A convergence result for a parallel algorithm for solving the Navier–Stokes equations, Comput. Math. Appl., Volume 35 (1998) no. 4, pp. 71-88

[3] J.R. Galo, PhD Thesis, Univ. de Sevilla, 2002

[4] J.R. Galo, I. Albarreal, M.C. Calzada, J.L. Cruz, E. Fernández-Cara, M. Marín, A parallel algorithm in time and space for the Navier–Stokes equations, in press

[5] J.R. Galo, I. Albarreal, M.C. Calzada, J.L. Cruz, E. Fernández-Cara, M. Marín, Analysis of a finite difference regularizing scheme with the same grid for the velocity and the pressure, in press

[6] U. Ghia; K.N. Ghia; C.T. Shin High-resolutions for incompressible flow using the Navier–Stokes equations and a multigrid method, J. Comput. Phys., Volume 48 (1982), pp. 387-411

[7] R. Glowinski Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984

[8] R. Glowinski Finite element methods for incompressible viscous flow, Handbooks of Numerical Analysis, vol. IX, North–Holland, Amsterdam, 2003, pp. 3-1176

[9] K. Kakuda; N. Tosaka; H. Kalo Petrov–Galerkin finite element approach using exponential functions for three-dimensional incompressible viscous flow problems, Finite Elements in Fluids (1993), pp. 204-213

[10] T. Lu; P. Neittaanmäki; X.C. Tai A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Math. Model. Num., Volume 26 (1992) no. 6, pp. 673-708

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