Abstract
We consider the time dependent Stokes equation on a finite time interval and on a uniform rectangular mesh, written in terms of velocity and pressure. In a parallel algorithm, based on a new direction splitting approach, the pressure equation is derived from a perturbed form of the continuity equation, in which the incompressibility constraint is penalized in a negative norm induced by the direction splitting. The scheme used in the algorithm is composed of: pressure prediction, velocity update, penalty step, and pressure correction. In order to achieve good parallel performance, the solution of the Poison problem for the pressure correction is replaced by solving a sequence of one-dimensional second order elliptic boundary value problems in each spatial direction. The parallel code was developed using MPI and tested on modern computer systems. The performed numerical tests illustrate the parallel efficiency, and the scalability, of the direction-splitting based algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. SIAM, Philadelphia (1999)
Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comp. 22, 745–762 (1968)
Guermond, J.L., Minev, P.: A new class of fractional step techniques for the incompressible Navier-Stokes equations using direction splitting. Comptes Rendus Mathematique 348(9-10), 581–585 (2010)
Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195, 6011–6054 (2006)
Guermond, J.L., Salgado, A.: A fractional step method based on a pressure poisson equation for incompressible flows with variable density. Comptes Rendus Mathematique 346(15-16), 913–918 (2008)
Guermond, J.L., Salgado, A.: A splitting method for incompressible flows with variable density based on a pressure Poisson equation. Journal of Computational Physics 228(8), 2834–2846 (2009)
Lirkov, I., Vutov, Y., Paprzycki, M., Ganzha, M.: Parallel Performance Evaluation of MIC(0) Preconditioning Algorithm for Voxel μFE Simulation. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds.) PPAM 2009, Part II. LNCS, vol. 6068, pp. 135–144. Springer, Heidelberg (2010)
Snir, M., Otto, S., Huss-Lederman, S., Walker, D., Dongarra, J.: MPI: The Complete Reference. Scientific and engineering computation series. The MIT Press, Cambridge (1997); second printing
Temam, R.: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. Arch. Rat. Mech. Anal. 33, 377–385 (1969)
Walker, D., Dongarra, J.: MPI: a standard Message Passing Interface. Supercomputer 63, 56–68 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lirkov, I., Paprzycki, M., Ganzha, M. (2012). Performance Analysis of Parallel Alternating Directions Algorithm for Time Dependent Problems. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2011. Lecture Notes in Computer Science, vol 7203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31464-3_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-31464-3_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31463-6
Online ISBN: 978-3-642-31464-3
eBook Packages: Computer ScienceComputer Science (R0)