Optimized Schwarz Waveform Relaxation Algorithms with Nonconforming Time Discretization for Coupling Convection-diffusion Problems with Discontinuous Coefficients

  • Eric Blayo
  • Laurence Halpern
  • Caroline Japhet
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 55)


We present and study an optimized SchwarzWaveform Relaxation algorithm for convection-diffusion problems with discontinuous coefficients. Such analysis is a first step towards the coupling of heterogeneous climatic models. The SWR algorithms are global in time, and thus allow for the use of non conforming space-time discretizations. They are therefore well adapted to coupling models with very different spatial and time scales, as in ocean-atmosphere coupling. As the cost per iteration can be very high, we introduce new transmission conditions in the algorithm which optimize the convergence speed. In order to get higher order schemes in time, we use in each subdomain a discontinuous Galerkin method for the time-discretization. We present numerical results to illustrate this approach, and we analyse numerically the time-discretization error.


Discontinuous Galerkin Method Transmission Condition High Order Scheme Time Grid Discontinuous Coefficient 
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  1. 1.
    D. Bennequin, M. J. Gander, and L. Halpern, Optimized Schwarz waveform relaxation methods for convection reaction diffusion problems, Tech. Rep. 24, Institut Galilée, Paris XIII, 2004.Google Scholar
  2. 2.
    M. J. Gander, L. Halpern, and M. Kern, A Schwarz Waveform Relaxation method for advection-diffusion-reaction problems with discontinuous coeffcients and non-matching grids, in Proceedings of the 16th International Conference on Domain Decomposition Methods, O. B. Widlund and D. E. Keyes, eds., Springer, 2006. these proceedings.Google Scholar
  3. 3.
    M. J. Gander, L. Halpern, and F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation, SIAM J. Numer. Anal., 41 (2003), pp. 1643–1681.zbMATHCrossRefGoogle Scholar
  4. 4.
    C. Johnson, Numerical Solutions of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1987.Google Scholar
  5. 5.
    V. Martin, An optimized Schwarz Waveform Relaxation method for the unsteady convection diffusion equation in two dimensions, Appl. Numer. Math., 52 (2005), pp. 401–428.zbMATHCrossRefGoogle Scholar
  6. 6.
    P. Pellerin, H. Ritchie, F. J. Saucier, F. Roy, S. Desjardins, M. Valin, and V. Lee, Impact of a two-way coupling between an atmospheric and an oceanice model over the gulf of St. Lawrence, Monthly Weather Review, 32 (2004), pp. 1379-1398.CrossRefGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Eric Blayo
    • 1
  • Laurence Halpern
    • 2
  • Caroline Japhet
    • 2
  1. 1.LMCUniversité Joseph FourierGrenoble Cedex 9France
  2. 2.LAGAUniversité Paris XIIIVilletaneuseFrance

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