Abstract
In a previous work (Angot et al. in J. Comput. Appl. Math. 226:228–245, 2009), some penalty–projection methods have been tested for the numerical analysis of the Navier-Stokes equations. The purpose of this study is to introduce a variant of the penalty–projection method which allows us to compute the solutions faster than by using the previous solver. This new variant combines dynamically and alternatively a penalty procedure and a projection procedure according to the size of the divergence of the velocity. In other words, this study aims to prove that it is possible to project the intermediate velocity, computed by the first step of the penalty–projection method, only if its divergence is larger than a specified threshold. Theoretical estimates for the new method are given, which are in accordance with the numerical results provided.
Similar content being viewed by others
References
Angot, Ph., Caltagirone, J.-P., Fabrie, P.: Vector penalty–projection methods for the solution of unsteady incompressible flows. In: 5th International Symposium on Finite Volumes for Complex Applications. Hermes Science (2008)
Angot, Ph., Févrière, C., Laminie, J., Poullet, P.: On the penalty–projection method for the Navier–Stokes equations with the MAC mesh. J. Comput. Appl. Math. 226, 228–245 (2009)
Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)
Févrière, C., Angot, Ph., Poullet, P.: A penalty-projection method using staggered grids for incompressible flows. Lect. Notes Comput. Sci. 4818, 192–200 (2008)
Fortin, M., Glowinski, R.: Augmented Lagrangian Methods, Applications to the Numerical Solution of Boundary Value Problems. North-Holland, Amsterdam (1983)
Griffith, B.E.: An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner. J. Comput. Phys. 228(20), 7565–7595 (2009)
Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006)
Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)
Jobelin, M., Lapuerta, C., Latché, J.-C., Angot, Ph., Piar, B.: A finite element penalty-projection method for incompressible flows. J. Comput. Phys. 217, 502–518 (2006)
Jobelin, M., Piar, B., Angot, Ph., Latché, J.-C.: Une méthode de pénalité–projection pour les écoulements dilatables. Eur. J. Comput. Mech. 17(4), 453–480 (2008)
Ni, M.-J.: Consistent projection methods for variable density incompressible Navier–Stokes equations with continuous surface forces on a rectangular collocated mesh. J. Comput. Phys. 228(18), 6938–6956 (2009)
Prohl, A.: On pressure approximation via projection methods for nonstationary incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 47(1), 158–180 (2008)
Shen, J.: On error estimates of some higher order projection and penalty-projection methods for Navier–Stokes equations. Numer. Math. 62, 49–73 (1992)
Shen, J.: On error estimates of projection methods for Navier–Stokes equations: second-order schemes. Math. Comput. 65(215), 1039–1065 (1996)
Sun, H., He, Y., Feng, X.: On error estimates of the pressure-correction projection methods for the time-dependent Navier–Stokes equations. Int. J. Numer. Anal. Model. 8(1), 70–85 (2011)
Temam, R.: Sur l’approximation par la méthode des pas fractionnaires. Arch. Ration. Mech. Anal. 32, 377–385 (1969)
Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1984). Revised version
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors are pleased to acknowledge the “Centre Commun de Calcul Intensif de l’Université des Antilles et de la Guyane” where the comptutational tests have been performed (see http://www.univ-ag.fr/c3i).
Rights and permissions
About this article
Cite this article
Laminie, J., Poullet, P. A Dynamic Penalty or Projection Method for Incompressible Fluids. J Sci Comput 50, 213–234 (2012). https://doi.org/10.1007/s10915-011-9480-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-011-9480-8