A Parallel Space-Time Finite Difference Solver for Periodic Solutions of the Shallow-Water Equation

  • Peter Arbenz
  • Andreas Hiltebrand
  • Dominik Obrist
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7204)


We investigate parallel algorithms for the solution of the shallow-water equation in a space-time framework. For periodic solutions, the discretized problem can be written as a large cyclic non-linear system of equations. This system of equations is solved with a Newton iteration which uses two levels of preconditioned GMRES solvers. The parallel performance of this algorithm is illustrated on a number of numerical experiments.


Periodic Solution Parallel Performance Shallow Water Equation Newton Iteration Newton Step 
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  1. 1.
    Chan, T.F.: An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Stat. Comput. 9, 766–771 (1988)zbMATHCrossRefGoogle Scholar
  2. 2.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Hiltebrand, A.: Parallel solution of time-periodic problems. Master thesis, ETH Zurich, Institute of Fluid Dynamics (March 2011)Google Scholar
  4. 4.
    Kevorkian, J.: Partial Differential Equations: Analytical Solution Techniques, 2nd edn. Springer, New York (2000)zbMATHGoogle Scholar
  5. 5.
    LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, Philadelphia (2007)zbMATHCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Obrist, D., Henniger, R., Arbenz, P.: Parallelization of the time integration for time-periodic flow problems. PAMM 10(1), 567–568 (2010)CrossRefGoogle Scholar
  8. 8.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)zbMATHCrossRefGoogle Scholar
  10. 10.
    Stoll, M., Wathen, A.: All-at-once solution of time-dependent PDE-constrained optimization problems. Technical Report 10/47, Oxford Centre for Collaborative Applied Mathematics, Oxford, England (2010)Google Scholar
  11. 11.
    The Trilinos Project Home Page,
  12. 12.
    Vreugdenhil, C.B.: Numerical Methods for Shallow-Water Flow. Kluwer, Dordrecht (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Peter Arbenz
    • 1
  • Andreas Hiltebrand
    • 2
  • Dominik Obrist
    • 3
  1. 1.Chair of Computational ScienceETH ZürichZürichSwitzerland
  2. 2.Seminar for Applied MathematicsETH ZürichZürichSwitzerland
  3. 3.Institute of Fluid DynamicsETH ZürichZürichSwitzerland

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