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Analysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation

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Abstract

The Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods are nonoverlapping spatial domain decomposition methods to solve evolution problems, while the parareal algorithm is in time parallel fashion. Based on the combinations of these space and time parallel strategies, we present and analyze two parareal algorithms based on the Dirichlet-Neumann and the Neumann-Neumann waveform relaxation method for the heat equation by choosing Dirichlet-Neumann/Neumann-Neumann waveform relaxation as two new kinds of fine propagators instead of the classical fine propagator. Both new proposed algorithms could be viewed as a space-time parallel algorithm, which increases the parallelism both in space and in time. We derive for the heat equation the convergence results for both algorithms in one spatial dimension. We also illustrate our theoretical results with numerical experiments finally.

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References

  1. Baffico, L., Bernard, S., Maday, Y., Turinici, G., Zérah, G.: Parallel-in-time molecular-dynamics simulations. Phys. Rev. E 66, 057701 (2002)

    Article  Google Scholar 

  2. Bal, G.: Parallelization in time of (stochastic) ordinary differential equations. Math. Meth. Anal Num. (submitted) (2003)

  3. Engblom, S.: Parallel in time simulation of multiscale stochastic chemical kinetics. Multiscale Model. Simul. 8(1), 46–68 (2009)

    Article  MathSciNet  Google Scholar 

  4. Fischer, P.F., Hecht, F., Maday, Y.: A parareal in time semi-implicit approximation of the Navier-Stokes equations. In: Kornhuber, R. et al. (eds.) Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, vol. 40, pp 433–440. Springer, Berlin (2005)

  5. Gander, M.J.: 50 years of time parallel time integration. In: Multiple Shooting and Time Domain Decomposition Methods, pp 69–113. Springer (2015)

  6. Gander, M.J., Hairer, E.: Nonlinear convergence analysis for the parareal algorithm. Lect. Notes Comput. Sci. Eng. 60, 45–56 (2008)

    Article  MathSciNet  Google Scholar 

  7. Gander, M.J., Halpern, L.: Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal. 45(2), 666–697 (2007)

    Article  MathSciNet  Google Scholar 

  8. Gander, M.J., Halpern, L., Nataf, F.: Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41(5), 1643–1681 (2003)

    Article  MathSciNet  Google Scholar 

  9. Gander, M.J., Jiang, Y.L., Li, R.J.: Parareal Schwarz waveform relaxation methods. In: Domain Decomposition Methods in Science and Engineering XX, pp 451–458. Springer (2013)

  10. Gander, M.J., Jiang, Y.L., Song, B.: A superlinear convergence estimate for the parareal Schwarz waveform relaxation algorithm. SIAM J. Sci. Comput. 41 (2), A1148–A1169 (2019)

    Article  MathSciNet  Google Scholar 

  11. Gander, M.J., Jiang, Y.L., Song, B., Zhang, H.: Analysis of two parareal algorithms for time-periodic problems. SIAM J. Sci. Comput. 35(5), A2393–A2415 (2013)

    Article  MathSciNet  Google Scholar 

  12. Gander, M.J., Kwok, F., Mandal, B.C.: Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems. Electron. Trans. Numer. Anal. 45, 424–456 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Gander, M.J., Kwok, F., Mandal, B.C.: Dirichlet-Neumann and Neumann-Neumann waveform relaxation for the wave equation. In: Domain Decomposition Methods in Science and Engineering XXII, pp 501–509. Springer (2016)

  14. Gander, M.J., Stuart, A.M.: Space-time continuous analysis of waveform relaxation for the heat equation. SIAM J. Sci. Comput. 19(6), 2014–2031 (1998)

    Article  MathSciNet  Google Scholar 

  15. Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)

    Article  MathSciNet  Google Scholar 

  16. Giladi, E., Keller, H.B.: Space-time domain decomposition for parabolic problems. Numer. Math. 93(2), 279–313 (2002)

    Article  MathSciNet  Google Scholar 

  17. Jiang, Y.L.: A general approach to waveform relaxation solutions of nonlinear differential-algebraic equations: the continuous-time and discrete-time cases. IEEE Trans. Circ. Syst. I: Regular Papers 51(9), 1770–1780 (2004)

    Article  MathSciNet  Google Scholar 

  18. Jiang, Y.L.: Waveform Relaxation Methods. Scientific Press, Beijing (2010)

    Google Scholar 

  19. Jiang, Y.L., Ding, X.L.: Waveform relaxation methods for fractional differential equations with the Caputo derivatives. J. Comput. Appl. Math. 238, 51–67 (2013)

    Article  MathSciNet  Google Scholar 

  20. Jiang, Y.L., Song, B.: Coupling parareal and Dirichlet-Neumann/Neumann-Neumann waveform relaxation methods for the heat equation. In: International Conference on Domain Decomposition Methods, pp 405–413. Springer (2017)

  21. Lelarasmee, E., Ruehli, A.E., Sangiovanni-Vincentelli, A.L.: The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Trans. Comput.-Aided Des. Integr. Circ. Syst. 1(3), 131–145 (1982)

    Article  Google Scholar 

  22. Lions, J.L., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDE’s. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 332(7), 661–668 (2001)

    Article  Google Scholar 

  23. Liu, J., Jiang, Y.L.: Waveform relaxation for reaction-diffusion equations. J. Comput. Appl. Mathe. 235(17), 5040–5055 (2011)

    Article  MathSciNet  Google Scholar 

  24. Liu, J., Jiang, Y.L.: A parareal algorithm based on waveform relaxation. Math. Comput. Simul. 82(11), 2167–2181 (2012)

    Article  MathSciNet  Google Scholar 

  25. Liu, J., Jiang, Y.L.: A parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations. J. Comput. Appl. Math. 236(17), 4245–4263 (2012)

    Article  MathSciNet  Google Scholar 

  26. Lubich, C., Ostermann, A.: Multi-grid dynamic iteration for parabolic equations. BIT Numer. Math. 27(2), 216–234 (1987)

    Article  Google Scholar 

  27. Maday, Y., Salomon, J., Turinici, G.: Monotonic parareal control for quantum systems. SIAM J. Numer. Anal. 45(6), 2468–2482 (2007)

    Article  MathSciNet  Google Scholar 

  28. Maday, Y., Turinici, G.: A parareal in time procedure for the control of partial differential equations. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 335(4), 387–392 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Maday, Y., Turinici, G.: Parallel in time algorithms for quantum control: parareal time discretization scheme. Int. J. Quant. Chem. 93(3), 223–228 (2003)

    Article  Google Scholar 

  30. Maday, Y., Turinici, G.: The parareal in time iterative solver: a further direction to parallel implementation. Lect. Notes Comput. Sci. Eng. 40, 441–448 (2005)

    Article  MathSciNet  Google Scholar 

  31. Song, B., Jiang, Y.L.: Analysis of a new parareal algorithm based on waveform relaxation method for time-periodic problems. Numer. Algor. 67(3), 599–622 (2014)

    Article  MathSciNet  Google Scholar 

  32. Song, B., Jiang, Y.L.: A new parareal waveform relaxation algorithm for time-periodic problems. Int. J. Comput. Math. 92(2), 377–393 (2015)

    Article  MathSciNet  Google Scholar 

  33. Staff, G.: Convergence and Stability of the Parareal Algorithm. Master’s thesis. Norwegian University of Science and Technology, Norway (2003)

    Google Scholar 

  34. Staff, G.A., Rønquist, E.M.: Stability of the parareal algorithm. Domain Decomposition Methods in Science and Engineering 40, 449–456 (2005)

    Article  MathSciNet  Google Scholar 

  35. Trindade, J.M.F., Pereira, J.C.F.: Parallel-in-time simulation of the unsteady Navier-Stokes equations for incompressible flow. Int. J. Numer. Methods Fluids 45 (10), 1123–1136 (2004)

    Article  Google Scholar 

  36. Vandewalle, S., Van de Velde, E.: Space-time concurrent multigrid waveform relaxation. Annals Numer. Math 1, 347–363 (1994)

  37. Wu, S.L.: Toward parallel coarse grid correction for the parareal algorithm. SIAM J. Sci. Comput. 40(3), A1446–A1472 (2018)

    Article  MathSciNet  Google Scholar 

  38. Wu, S.L., Al-Khaleel, M.: Convergence analysis of the Neumann-Neumann waveform relaxation method for time-fractional RC circuits. Simul. Model. Pract. Theory 64, 43–56 (2016)

    Article  Google Scholar 

  39. Wu, S.L., Zhou, T.: Fast parareal iterations for fractional diffusion equations. J. Comput. Phys. 329, 210–226 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We would like to thank the anonymous referee for his valuable comments.

Funding

This work was supported by the Natural Science Foundation of China (NSFC) under grants 11801449, 11871393 and 11871400; the key project of the International Science and Technology Cooperation Program of Shaanxi Research & Development Plan under grant 2019KWZ-08; the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2019JQ-617) and the Fundamental Research Funds for the Central Universities under grant G2018KY0306.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China

    Bo Song & Xiaolong Wang

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Yao-Lin Jiang

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  1. Bo Song
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  2. Yao-Lin Jiang
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  3. Xiaolong Wang
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Correspondence to Bo Song.

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Song, B., Jiang, YL. & Wang, X. Analysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation. Numer Algor 86, 1685–1703 (2021). https://doi.org/10.1007/s11075-020-00949-y

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