A Parallel Multigrid Solver for Time-Periodic Incompressible Navier–Stokes Equations in 3D

  • Pietro BenedusiEmail author
  • Daniel Hupp
  • Peter Arbenz
  • Rolf Krause
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 112)


We present a parallel and efficient multilevel solution method for the nonlinear systems arising from the discretization of Navier–Stokes (N-S) equations with finite differences. In particular we study the incompressible, unsteady N-S equations with periodic boundary condition in time. A sequential time integration limits the parallelism of the solver to the spatial variables and can therefore be an obstacle to parallel scalability. Time periodicity allows for a space-time discretization, which adds time as an additional direction for parallelism and thus can improve parallel scalability. To achieve fast convergence, we used a space-time multigrid algorithm with a SCGS smoothing procedure (symmetrical coupled Gauss–Seidel, a.k.a. box smoothing). This technique, proposed by Vanka (J Comput Phys 65:138–156, 1986), for the steady viscous incompressible Navier–Stokes equations is extended to the unsteady case and its properties are studied using local Fourier analysis. We used numerical experiments to analyze the scalability and the convergence of the solver, focusing on the case of a pulsatile flow.


Pulsatile Flow Stagger Grid Multigrid Algorithm Defect Correction Multigrid Solver 
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Pietro Benedusi and Rolf Krause would like to thank the Swiss National Science Foundation (SNF) and the Deutsche Forschungsgemeinschaft for supporting this work in the framework of the project “ExaSolvers – Extreme Scale Solvers for Coupled Systems ”, SNF project number 145271, within the DFG-Priority Research Program 1684 “SPPEXA- Software for Exascale Computing”


  1. 1.
    P. Benedusi, A parallel multigrid solver for time-periodic incompressible Navier–Stokes equations. Master thesis, USI Lugano, ICS (2015)Google Scholar
  2. 2.
    S.P. Vanka, Block-implicit multigrid solution of Navier–Stokes equations in primitive variables. J. Comput. Phys. 65, 138–156 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    I.V. Pivkin, P.D. Richardson, D.H. Laidlaw, G.E. Karniadakis, Combined effects of pulsatile flow and dynamic curvature on wall shear stress in a coronary artery bifurcation model. J. Biomech. 38, 1283–1290 (2015)CrossRefGoogle Scholar
  4. 4.
    M. Mehrabi, S. Setayeshi, Computational fluid dynamics analysis of pulsatile blood flow behavior in modelled stenosed vessels with different severities. Math. Probl. Eng. 2012, 13 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J.B. Grotberg, Pulmonary flow and transport phenomena. Annu. Rev. Fluid Mech. 26, 529–571 (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    B. Koren, Multigrid and defect correction for the steady Navier–Stokes equations. J. Comput. Phys. 87, 25–46 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    W. Ming, C. Long, Multigrid methods for the stokes equations using distributive Gauss–Seidel relaxations based on the least squares commutator. J. Sci. Comput. 56 (2013)Google Scholar
  8. 8.
    A. Brandt, Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    L.B. Zhang, Box-line relaxation schemes for solving the steady incompressible Navier–Sotkes equaitons using second order upwind differincing. J. Comput. Math. 13, 32–39 (1991)Google Scholar
  10. 10.
    S. Sivaloganathan, The use of local mode analysis in the design and comparison of multigrid methods. Comput. Phys. Comm. 65, 246–252 (1991)CrossRefzbMATHGoogle Scholar
  11. 11.
    Heroux et al., An overview of the trilinos project. ACM Trans. Math. Softw. 31, 397–423 (2005)Google Scholar
  12. 12.
    R. Henniger, D. Obrist, L. Kleiser, High-order accurate solution of the incompressible Navier–Stokes equations on massively parallel computers. J. Comput. Phys. 229, 3543–3572 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. Linden et al., Multigrid for the steady-state incompressible Navier–Stokes equations: a survey, in 11th International Conference on Numerical Methods in Fluid Dynamics (Springer, Berlin/Heidelberg, 1989)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Pietro Benedusi
    • 1
    Email author
  • Daniel Hupp
    • 2
  • Peter Arbenz
    • 2
  • Rolf Krause
    • 1
  1. 1.Institute of Computational Science, USILuganoSwitzerland
  2. 2.Computer Science Department, ETHZZürichSwitzerland

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