Time Parallelization for Nonlinear Problems Based on Diagonalization

  • Martin J. Gander
  • Laurence HalpernEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 116)


The direct time parallelization method based on diagonalization is only applicable to linear problems. We propose here a new method based on diagonalization which permits the direct parallelization in time of a Newton iteration that works simultaneously over several time steps. We first explain the method for a scalar model problem, and then give a formulation for a nonlinear partial differential equation based on tensorization. We illustrate the methods with numerical experiments.


  1. D. Bennequin, M.J. Gander, L. Halpern, A homographic best approximation problem with application to optimized Schwarz waveform relaxation. Math. Comput. 78 (265), 185–223 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. M. Bjørhus, On domain decomposition, subdomain iteration and waveform relaxation. PhD thesis, University of Trondheim, Norway (1995)Google Scholar
  3. P. Chartier, B. Philippe, A parallel shooting technique for solving dissipative ODEs. Computing 51, 209–236 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. A.J. Christlieb, C.B. Macdonald, B.W. Ong, Parallel high-order integrators. SIAM J. Sci. Comput. 32 (2), 818–835 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. M. Emmett, M.L. Minion, Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci 7 (1), 105–132 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. M.J. Gander, 50 years of time parallel time integration, in Multiple Shooting and Time Domain Decomposition Methods (Springer, Berlin, 2015), pp. 69–113CrossRefzbMATHGoogle Scholar
  7. M.J. Gander, S. Güttel, Paraexp: a parallel integrator for linear initial-value problems. SIAM J. Sci. Comput. 35 (2), C123–C142 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. M.J. Gander, E. Hairer, Nonlinear convergence analysis for the parareal algorithm. in Domain Decomposition Methods in Science and Engineering XVII, vol. 60 (Springer, Berlin, 2008), pp. 45–56Google Scholar
  9. M.J. Gander, L. Halpern, Absorbing boundary conditions for the wave equation and parallel computing. Math. Comput. 74, 153–176 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. M.J. Gander, L. Halpern, Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal. 45 (2), 666–697 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. M.J. Gander, M. Neumüller, Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. 38 (4), A2173–A2208 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. M.J. Gander, A.M. Stuart, Space-time continuous analysis of waveform relaxation for the heat equation. SIAM J. Sci. Comput. 19 (6), 2014–2031 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. M.J. Gander, S. Vandewalle, Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29 (2), 556–578 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. M.J. Gander, L. Halpern, F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41 (5), 1643–1681 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. M.J. Gander, L. Halpern, J. Ryan, T.T.B. Tran, A direct solver for time parallelization, in 22nd International Conference of Domain Decomposition Methods (Springer, Berlin, 2014)Google Scholar
  16. M.J. Gander, L. Halpern, J. Rannou, J. Ryan, A direct solver for time parallelization of the wave equation. (2016a, in preparation)Google Scholar
  17. M.J. Gander, F. Kwok, B. Mandal, Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems. Electron. Trans. Numer. Anal. 45, 424–456 (2016b)MathSciNetzbMATHGoogle Scholar
  18. E. Giladi, H.B. Keller, Space time domain decomposition for parabolic problems. Numer. Math. 93 (2), 279–313 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. W. Hackbusch, Parabolic multi-grid methods, in Computing Methods in Applied Sciences and Engineering, VI, (North-Holland, Amsterdam, 1984), pp. 189–197Google Scholar
  20. G. Horton, S. Vandewalle, A space-time multigrid method for parabolic partial differential equations. SIAM J. Sci. Comput. 16 (4), 848–864 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. F. Kwok, Neumann–Neumann waveform relaxation for the time-dependent heat equation, in Domain Decomposition Methods in Science and Engineering XXI (Springer, Berlin, 2014), pp. 189–198Google Scholar
  22. J.L. Lions, Y. Maday, G. Turinici, A parareal in time discretization of PDE’s. C.R. Acad. Sci. Paris Ser. I 332, 661–668 (2001)Google Scholar
  23. Y. Maday, E.M. Rønquist, Parallelization in time through tensor-product space-time solvers. C. R. Math. Acad. Sci. Paris 346 (1–2), 113–118 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. B. Mandal, A time-dependent Dirichlet-Neumann method for the heat equation, in Domain Decomposition Methods in Science and Engineering, DD21 (Springer, Berlin, 2014)Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of GenevaGenevaSwitzerland
  2. 2.University Paris 13ParisFrance

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