Skip to main content
SpringerLink
Log in

Analysis of a New Accelerated Waveform Relaxation Method Based on the Time-Parallel Algorithm

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we propose a new accelerated waveform relaxation (WR) method based on a time-parallel algorithm to solve the general system of ordinary differential equations (ODEs). It is well known that the WR method decouples or linearizes large-scale complex systems into simple subsystems, which in most cases can be computed in parallel in each iteration. To accelerate the calculation of the WR iteration, we apply a time-parallel approach: the Parareal algorithm, to solve the subsystems in each iteration. It can be thought of as a kind of space-time parallel method. According to different WR types, we present convergence analysis of the accelerated WR methods for the time-continuous case and for the time-discrete case with different discrete schemes. Besides, the speedup analysis of the proposed algorithms is also provided. Finally, numerical experiments are carried out to verify the effectiveness of the theoretical works.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Data Availability

All data generated or analysed during the current study are included in this paper.

References

  1. Al-Khaleel, M., Gander, M.J., Ruehli, A.E.: Optimization of transmission conditions in waveform relaxation techniques for RC circuits. SIAM J. Numer. Anal. 52, 1076–1101 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chaudhry, J.H., Estep, D., Tavener, S., Carey, V., Sandelin, J.: A posteriori error analysis of two-stage computation methods with application to efficient discretization and the parareal algorithm. SIAM J. Numer. Anal. 54, 2974–3002 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Du, X.H., Sarkis, M., Schaerer, C.F., Szyld, D.B.: Inexact and truncated parareal in-time Krylov subspace methods for parabolic optimal control problems. Electron. Trans. Numer. Anal. 40, 36–57 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Eager, D.L., Zahorjan, J., Lazowska, E.D.: Speedup versus efficiency in parallel systems. IEEE Trans. Comput. 38, 408–423 (1989)

    Article  Google Scholar 

  5. Erdman, J., Rose, J.: Newton waveform relaxation techniques for tightly coupled systems. IEEE Trans. CAD IC Syst. 11, 598–606 (1992)

    Article  Google Scholar 

  6. Fu, H., Wang, H.: A preconditioned fast parareal finite difference method for space-time fractional partial differential equation. J. Sci. Comput. 78, 1724–1743 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gu, C.: QLMOR: A projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems. IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 30, 1307-1320(2011)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Gander, M.J., Hairer, E.: Nonlinear convergence analysis for the parareal algorithm. Domain Decompos. Methods Sci. Eng. 60, 45–56 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Gander, M.J., Wu, S.L.: A diagonalization-based parareal algorithm for dissipative and wave propagation problems. SIAM J. Numer. Anal. 58, 2981–3009 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Horton, G., Vandewalle, S.: A spacetime multigrid method for parabolic PDEs. SIAM J. Sci. Comput. 16, 848–864 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press (1985)

  15. Jeltsch, R., Pohl, B.: Waveform relaxation with overlapping splittings. SIAM J. Sci. Comput. 16, 40–49 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Janssen, J.: Acceleration of waveform relaxation methods for linear ordinary and partial differential equations. Ph.D. Thesis, Department of Computer Science, Katholieke Universiteit Leuven (1997)

  17. Janssen, J., Vandewalle, S.: Multigrid waveform relaxation of spatial finite element meshes: the continuous-time case. SIAM J. Numer. Anal. 33, 456–474 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiang, Y.L.: On time-domain simulation of lossless transmission lines with nonlinear terminations. SIAM J. Numer. Anal. 42, 1018–1031 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jiang, Y.L., Chen, R., Wing, O.: Periodic waveform relaxation of nonlinear dynamic systems by quasi-linearization. IEEE Trans. Circuits Syst. I(50), 589–593 (2003)

    Article  Google Scholar 

  20. Jiang, J.L.: Windowing waveform relaxation of initial value problem. Acta Math. Appl. Sin. Engl. Ser. 22, 575–588 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jiang, Y.L., Miao, Z.: Waveform relaxation of partial differential equations. Numer. Algor 79, 1087–1106 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lelarasmee, E., Ruehli, A., Sangiovanni-Vincentelli, A.: The waveform relaxation method for the time-domain analysis of large scale integrated circuits. IEEE T. Comput. Aid. D. 1, 131–145 (1982)

    Article  Google Scholar 

  23. Leimkuhler, B.: Timestep acceleration of waveform relaxation. SIAM J. Numer. Anal. 35, 31–50 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lions, J.L., Maday, Y., Turinici, G.: A “parareal’’ in time discretization of PDE’s. C. R. Acad. Sci. Paris Ser. I Math. 332, 661–668 (2001)

    Article  MATH  Google Scholar 

  25. Li, J., Jiang, Y.L., Miao, Z.: A parareal approach of semi-linear parabolic equations based on general waveform relaxation. Numer. Meth. Part. Dffer. Equ. 35, 2017–2043 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Monge, A., Birken, P.: A multirate Neumann–Neumann waveform relaxation method for heterogeneous coupled heat equations. SIAM J. Sci. Comput. 41, 86–105 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mathew, T.R., Sarkis, M., Schaerer, C.E.: Analysis of block parareal preconditioners for parabolic optimal controal problems. SIAM J. Sci. Comput. 32, 1180–1200 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nevanlinna, O.: Remarks on Picard Lindelöf iteration. BIT 29, 328–346 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ruehli, A.E., Johnson, T.A.: Circuit Analysis Computing by Waveform Relaxation, in Encyclopedia of Electrical and Electronics Engineering. Wiley, New York (1999)

    Google Scholar 

  32. Staff, G.A., Ronquist, E.M.: Stability of the parareal algorithm. Lect. Notes Comput. Sci. Eng. 40, 449–456 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Taasan, S., Zhang, H.: On the multigrid waveform relaxation method. SIAM J. Sci. Comput. 16, A1092–A1104 (1995)

    Article  MathSciNet  Google Scholar 

  34. Vandewalle, S., Piessens, R.: On dynamic iteration methods for solving time-periodic differential equations. SIAM J. Numer. Anal. 30, 286–303 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are sincerely thankful to the anonymous reviewer for the valuable suggestions and comments to improve our work. This work was supported by the Natural Science Foundation of China (NSFC) under grant 12271426, the Key Research and Development Projects of Shaanxi Province under grant 2023-YBSF-399.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Jun Li & Yaolin Jiang

Authors
  1. Jun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yaolin Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yaolin Jiang.

Ethics declarations

Conflict of interest

The authors have not disclosed any conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, J., Jiang, Y. Analysis of a New Accelerated Waveform Relaxation Method Based on the Time-Parallel Algorithm. J Sci Comput 96, 68 (2023). https://doi.org/10.1007/s10915-023-02285-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02285-4

Keywords

Search

Navigation