An Adaptive Parallel-in-Time Method with Application to a Membrane Problem

  • Noha Makhoul Karam
  • Nabil Nassif
  • Jocelyne Erhel
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 98)


In a previous work (Nassif et al., In: V.A. al (ed.) ICCS 06. Computer Science, pp. 148–155. Springer, New York, 2006), we introduced an approach for solving the initial value problem \(\frac{\mathit{dY}} {\mathit{dt}} = F(t,Y ),\,Y (0) = Y _{0}\) in a time-parallel way. The main feature of the method is its capacity to automatically generate a non-regular time grid, adapted to the behavior of the solution. Parallel integration is made possible by introducing a “shooting function” that partitions the problem into a sequence of shooting value problems, each defined on a time slice of the coarse grid. After rescaling the variables on each slice, a prediction data model that permits accurate predictions of the solution at the beginning of every slice, leads to an Adaptive Parallel Time Integration (APTI) algorithm. In this paper, the method is applied to a membrane problem having oscillatory and unbounded solutions on (0, ).


Coarse Grid Waveform Relaxation Membrane Problem Multiple Shoot Method Asymptotic Similarity 
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.


  1. 1.
    Bal, G., Wu, Q.: Symplectic parareal. In: Bercovier, M., Gander, M., Kornhuber, R., Widlund, O. (eds.) DD08. Lecture Notes in Computational Science and Engineering, pp. 189–202. Springer, Berlin (2008)Google Scholar
  2. 2.
    Chartier, P., Philippe, B.: A parallel shooting technique for solving dissipative ode’s. Computing 51, 209–236 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Erhel, J., Rault, S.: Algorithme parallèle pour le calcul d’orbites. Techniques et Sciences Informatiques 19, 649–673 (2000)Google Scholar
  4. 4.
    Farhat, C., Chandesris, M.: Time-decomposed parallel time-integrators. Int. J. Numer. Methods Eng. 58, 1397–1434 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Lions, J., Maday, Y., Turinici, G.: Résolution d’edp par un schéma en temps “pararéel”. C. R. Acad. Sci. Paris 332, 661–668 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Maday, Y., Bal, G.: A parareal time discretization for non-linear pde’s with application to the pricing of an american put. In: L.F. Pavarino and A. Toselli (ed.) DD02. Computer Science, pp. 189–202. Springer, Berlin (2002)Google Scholar
  7. 7.
    Makhoul-Karam, N.: Time-slicing, rescaling & ratio-based parallel time integration. TEL (2012).
  8. 8.
    Nassif, N., Fayad, D., Cortas, M.: Sliced-time computations with rescaling for blowing-up solutions to init. val. pbs. In: V.S. Sunderam et al. (ed.) ICCS 05. Computer Science, pp. 58–65. Springer, New York (2005)Google Scholar
  9. 9.
    Nassif, N., Makhoul-Karam, N., Soukiassian, Y.: A new approach for solving evolution problems in time-parallel way. In: V.N. Alexandrov et al. (ed.) ICCS 06. Computer Science, pp. 148–155. Springer, New York (2006)Google Scholar
  10. 10.
    Nassif, N., Makhoul-Karam, N., Soukiassian, Y.: Computation of blowing-up solutions for second-order differential equations using re-scaling techniques. JCAM 227, 185–195 (2009)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Nassif, N., Makhoul-Karam, N., Erhel, J.: Globally adaptive explicit numerical methods for exploding systems of ordinary differential equations. APNUM (2011).
  12. 12.
    Nievergelt, J.: Parallel methods for integration of ode’s. Commun. ACM 7, 731–733 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Souplet, P.: Critical exponents, special large-time behavior and oscillatory blow-up in nonlinear ode’s. Differ. Integral Equ. 11, 147–167 (1998)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Noha Makhoul Karam
    • 1
  • Nabil Nassif
    • 2
  • Jocelyne Erhel
    • 3
  1. 1.Université Saint JosephBeyrouthLebanon
  2. 2.American University of BeirutBeirutLebanon
  3. 3.INRIARennesFrance

Personalised recommendations