Performance Analysis of Parallel Alternating Directions Algorithm for Time Dependent Problems

  • Ivan Lirkov
  • Marcin Paprzycki
  • Maria Ganzha
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7203)


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.


Execution Time Parallel Performance Pressure Correction Incompressibility Constraint Pressure Poisson Equation 
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  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comp. 22, 745–762 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195, 6011–6054 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    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 Google Scholar
  9. 9.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Walker, D., Dongarra, J.: MPI: a standard Message Passing Interface. Supercomputer 63, 56–68 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ivan Lirkov
    • 1
  • Marcin Paprzycki
    • 2
  • Maria Ganzha
    • 2
  1. 1.Institute of Information and Communication TechnologiesBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Systems Research InstitutePolish Academy of SciencesWarsawPoland

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