Parallel-in-Space-and-Time Simulation of the Three-Dimensional, Unsteady Navier-Stokes Equations for Incompressible Flow

  • Roberto Croce
  • Daniel Ruprecht
  • Rolf Krause
Conference paper


In this paper we combine the Parareal parallel-in-time method together with spatial parallelization and investigate this space-time parallel scheme by means of solving the three-dimensional incompressible Navier-Stokes equations. Parallelization of time stepping provides a new direction of parallelization and allows to employ additional cores to further speed up simulations after spatial parallelization has saturated. We report on numerical experiments performed on a Cray XE6, simulating a driven cavity flow with and without obstacles. Distributed memory parallelization is used in both space and time, featuring up to 2,048 cores in total. It is confirmed that the space-time-parallel method can provide speedup beyond the saturation of the spatial parallelization.


Domain Decomposition Spatial Parallelization Total Speedup Drive Cavity Flow Distribute Memory Parallelization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This research is funded by the Swiss “High Performance and High Productivity Computing” initiative HP2C. Computational resources were provided by the Swiss National Supercomputing Centre CSCS.


  1. 1.
    Chorin, A.J.: Numerical solution of the Navier Stokes equations. Math. Comput. 22(104), 745–762 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Croce, R., Engel, M., Griebel, M., Klitz, M.: NaSt3DGP – a Parallel 3D Flow Solver.
  3. 3.
    Emmett, M., Minion, M.L.: Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci. 7, 105–132 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Farhat, C., Chandesris, M.: Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications. Int. J. Numer. Methods Eng. 58, 1397–1434 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fischer, P.F., Hecht, F., Maday, Y.: A parareal in time semi-implicit approximation of the Navier-Stokes equations. In: Kornhuber, R., et al. (eds.) Domain Decomposition Methods in Science and Engineering. LNCSE, vol. 40, pp. 433–440. Springer, Berlin (2005)CrossRefGoogle Scholar
  6. 6.
    Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gaskell, P., Lau, A.: Curvature-compensated convective transport: SMART a new boundedness-preserving transport algorithm. Int. J. Numer. Methods Fluids 8, 617–641 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Griebel, M., Dornseifer, T., Neunhoeffer, T.: Numerical Simulation in Fluid Dynamics, a Practical Introduction. SIAM, Philadelphia (1998)CrossRefGoogle Scholar
  9. 9.
    Leonard, B.: A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Methods Appl. Mech. Eng. 19, 59–98 (1979)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lions, J.L., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDE’s. C. R. Acad. Sci. – Ser. I – Math. 332, 661–668 (2001)Google Scholar
  11. 11.
    Minion, M.L.: A hybrid parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(2), 265–301 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ruprecht, D., Krause, R.: Explicit parallel-in-time integration of a linear acoustic-advection system. Comput. Fluids 59, 72–83 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Temam, R.: Sur l’approximation de la solution des equations de Navier-Stokes par la méthode des pas fractionnaires II. Arch. Ration. Mech. Anal. 33, 377–385 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Trindade, J.M.F., Pereira, J.C.F.: Parallel-in-time simulation of the unsteady Navier-Stokes equations for incompressible flow. Int. J. Numer. Methods. Fluids 45, 1123–1136 (2004)CrossRefzbMATHGoogle Scholar
  15. 15.
    Trindade, J.M.F., Pereira, J.C.F.: Parallel-in-time simulation of two-dimensional, unsteady, incompressible laminar flows. Numer. Heat Trans., Part B 50, 25–40 (2006)Google Scholar
  16. 16.
    van der Vorst, H.: Bi-CGStab: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631 (1992)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institute of Computational ScienceLuganoSwitzerland

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