BIT Numerical Mathematics

, Volume 58, Issue 3, pp 807–834 | Cite as

A parallel spectral deferred correction method for first-order evolution problems

  • Shuai Zhu
  • Shilie Weng


This paper investigates a novel parallel technique based on the spectral deferred correction (SDC) method and a compensation step for solving first-order evolution problems, and we call it para-SDC method for convenience. The standard SDC method is used in parallel with a rough initial guess and a Picard integral equation with high precision initial condition is acted as a compensator. The goal of this paper is to show how these processes can be parallelized and how to improve the efficiency. During the SDC step an implicit or semi-implicit method can be used for stiff problems which is always time-consuming, therefore that’s why we do this procedure in parallel. Due to a better initial condition of parallel intervals after the SDC step, the goal of compensation step is to get a better approximation and also avoid of solving an implicit problem again. During the compensation step an explicit Picard scheme is taken based on the numerical integration with polynomial interpolation on Gauss Radau II nodes, which is almost no time consumption, obviously, that’s why we do this procedure in serial. The convergency analysis and the parallel efficiency of our method are also discussed. Several numerical experiments and an application for simulation Allen–Cahn equation are presented to show the accuracy, stability, convergence order and efficiency features of para-SDC method.


Spectral deferred correction Parareal algorithm Stiff system Preconditioning 

Mathematics Subject Classification

34K28 74H15 74Q10 74S05 



The authors thank Ph.D. Huashan Sheng for his critical support and technical assistance.


  1. 1.
    Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)CrossRefGoogle Scholar
  2. 2.
    Amodio, P., Brugnano, L.: Parallel Solution in Time of ODEs: Some Achievements and Perspectives. Elsevier, Amsterdam (2009)zbMATHGoogle Scholar
  3. 3.
    Bal, G., Maday, Y.: A Parareal Time Discretization for Non-Linear PDEs with Application to the Pricing of an American Put. Springer, Berlin (2002)zbMATHGoogle Scholar
  4. 4.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  5. 5.
    Bu, S., Lee, J.Y.: An enhanced parareal algorithm based on the deferred correction methods for a stiff system. J. Comput. Appl. Math. 255(285), 297–305 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Burrage, K.: Parallel methods for ODEs. Adv. Comput. Math. 7(1), 1–31 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chehab, J.P., Petcu, M.: Parallel matrix function evaluation via initial value ODE modeling. Comput. Math. Appl. 72(1), 76–91 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dai, X., Bris, C.L., Maday, F.L.Y.: Symmetric parareal algorithms for Hamiltonian systems. ESAIM Math. Model. Numer. Anal. 47(3), 717–742 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT Numer. Math. 40(2), 241–266 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Emmett, M., Minion, M.L.: Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci. 7(1), 105–132 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Farhat, C., Chandesris, M.: Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid–structure applications. Int. J. Numer. Methods Eng. 58(9), 1397–1434 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Farhat, C., Cortial, J., Dastillung, C., Bavestrello, H.: Time-parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses. Int. J. Numer. Meth. Eng. 67(5), 697–724 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Feng, X., Tang, T., Yang, J.: Long time numerical simulations for phase-field problems using p-adaptive spectral deferred correction methods. SIAM J. Sci. Comput. 37(1), A271–A294 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fischer, P.F., Hecht, F., Maday, Y.: A Parareal in Time Semi-implicit Approximation of the Navier–Stokes Equations. Springer, Berlin (2005)CrossRefGoogle Scholar
  15. 15.
    Gander, M.J., Gttel, S.: Paraexp: a parallel integrator for linear initial-value problems. SIAM J. Sci. Comput. 35(2), C123–C142 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gander, M.J., Vandewalle, S.: Analysis of the parareal timeparallel time integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hagstrom, T., Zhou, R.: On the spectral deferred correction of splitting methods for initial value problems. Commun. Appl. Math. Comput. Sci. 1(1), 169–205 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lions, J., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDE’s. Comptes Rendus delAcademie des Sciences-Series I -Mathematics 332(7), 661–668 (2001)zbMATHGoogle Scholar
  19. 19.
    Liu, J., Wang, X.: An assessment of the differential quadrature time integration scheme for nonlinear dynamic equations. J. Sound Vib. 314(1), 246–253 (2008)CrossRefGoogle Scholar
  20. 20.
    Maday, Y., Turinici, G.: The Parareal in Time Iterative Solver: A Further Direction to Parallel Implementation. Springer, Berlin (2005)zbMATHGoogle Scholar
  21. 21.
    Minion, M.L., Williams, S.A.: Parareal and spectral deferred corrections. In: AIP Conference Proceedings, pp. 388–391 (2008)Google Scholar
  22. 22.
    Minion, M.L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 2127–2157 (2002)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Minion, M.: A hybrid parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(5), 265–301 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Parter, S.V.: On the Legendre–Gauss—Lobatto points and weights. J. Sci. Comput. 14(4), 347–355 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer, New York (2011)zbMATHGoogle Scholar
  26. 26.
    Shin, J., Park, S.K., Kim, J.: A hybrid FEM for solving the Allen–Cahn equation. Appl. Math. Comput. 244(2), 606–612 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Wood, W.: Practical Time-Stepping Schemes. Oxford University Press, Oxford (1990)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mechanical EngeeringShanghai Jiao Tong UniversityShanghaiChina

Personalised recommendations