Advertisement

On the Convergence and the Stability of the Parareal Algorithm to Solve Partial Differential Equations

  • Guillaume Bal
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)

Summary

After stating an abstract convergence result for the parareal algorithm used in the parallelization in time of general partial differential equations, we analyze the stability and convergence properties of the algorithm for equations with constant coefficients. We show that suitably damping coarse schemes ensure unconditional stability of the parareal algorithm and analyze how the regularity of the initial condition influences convergence in the absence of sufficient damping.

Keywords

Parabolic Equation Spatial Discretization Domain Decomposition Method Parareal Scheme Unconditional Stability 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Baffico, S. Bernard, T. Maday, G. Turinici, and G. Zérah. Parallel-in-time molecular-dynamics simulations. Phys. Rev. E, 66:057701, 2002.CrossRefGoogle Scholar
  2. G. Bal. Parallelization in time of (stochastic) ordinary differential equations. Preprint; www.columbia.edu/~gb2030/PAPERS/ParTimeSODE.ps, 2003.Google Scholar
  3. G. Bal and Y. Maday. A “parareal” time discretization for non-linear PDE's with application to the pricing of an american put. Recent developments in domain decomposition methods (Zürich, 2001), Lect. Notes Comput. Sci. Eng., Springer, Berlin, 23:189–202, 2002.Google Scholar
  4. C. Farhat and M. Chandesris. Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications. Int. J. Numer. Meth. Engng., 58(9):1397–1434, 2003.MathSciNetCrossRefGoogle Scholar
  5. J.-L. Lions, Y. Maday, and G. Turinici. Résolution d'EDP par un schéma en temps “pararéel”. C.R.A.S. Sér. I Math., 332(7):661–668, 2000.MathSciNetGoogle Scholar
  6. Y. Maday and G. Turinici. A parareal in time procedure for the control of partial differential equations. C.R.A.S. Sér. I Math., 335:387–391, 2002.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Guillaume Bal
    • 1
  1. 1.APAMColumbia UniversityUSA

Personalised recommendations