A Parallel Solver for the Time-Periodic Navier–Stokes Equations

  • Peter Arbenz
  • Daniel Hupp
  • Dominik Obrist
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8385)


We investigate parallel algorithms for the solution of the Navier–Stokes equations in space-time. For periodic solutions, the discretized problem can be written as a large non-linear system of equations. This system of equations is solved by a Newton iteration. The Newton correction is computed using a preconditioned GMRES solver. The parallel performance of the algorithm is illustrated.


Time-periodic Navier-Stokes Space-time parallelism Nonlinear systems of equations Newton iteration GMRES 


  1. 1.
    Arbenz, P., Hiltebrand, A., Obrist, D.: A parallel space-time finite difference solver for periodic solutions of the shallow-water equation. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds.) PPAM 2011, Part II. LNCS, vol. 7204, pp. 302–312. Springer, Heidelberg (2012) Google Scholar
  2. 2.
    Christlieb, A.J., Haynes, R.D., Ong, B.W.: A parallel space-time algorithm. SIAM J. Sci. Comput. 34(5), C233–C248 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Henniger, R., Obrist, D., Kleiser, L.: High-order accurate solution of the incompressible Navier-Stokes equations on massively parallel computers. J. Comput. Phys. 229(10), 3543–3572 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Thornquist, H.K., Tuminaro, R.S., Willenbring, J.M., Williams, A., Stanley, K.S.: An overview of the Trilinos project. ACM Trans. Math. Softw. 31(3), 397–423 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Hupp, D.: A parallel space-time solver for Navier-Stokes. Master thesis, ETH Zurich, Curriculum Computational Science and Engineering., May 2013
  6. 6.
    LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, Philadelphia (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Lions, J.-L., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDE’s. C. R. Math. Acad. Sci. Paris 332(7), 661–668 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Mohd-Yusof, J.: Combined immersed-boundary/B-spline methods for simulations of flow in complex geometries. Annual Research Briefs, pp. 317–327. NASA Ames/Stanford Univ., Center for Turbulence Research. (1997)
  9. 9.
    Obrist, D., Henniger, R., Arbenz, P.: Parallelization of the time integration for time-periodic flow problems. PAMM 10(1), 567–568 (2010)CrossRefGoogle Scholar
  10. 10.
    Pawlowski, R.P., Shadid, J.N., Simonis, J.P., Walker, H.F.: Globalization techniques for Newton-Krylov methods and applications to the fully coupled solution of the Navier-Stokes equations. SIAM Rev. 48(4), 700–721 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Saad, Y.: IIUT: a dual threshold incomplete LU factorization. Numer. Linear Algebra Appl. 1(4), 387–402 (1994)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Computer Science DepartmentETH ZürichZürichSwitzerland
  2. 2.Institute of Fluid DynamicsETH ZürichZürichSwitzerland

Personalised recommendations