Skip to main content
SpringerLink
Log in

Long-Term Stability of Symmetric Partitioned Linear Multistep Methods

  • Chapter
  • First Online:
Current Challenges in Stability Issues for Numerical Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2082))

Abstract

Long-time integration of Hamiltonian systems is an important issue in many applications—for example the planetary motion in astronomy or simulations in molecular dynamics. Symplectic and symmetric one-step methods are known to have favorable numerical features like near energy preservation over long times and at most linear error growth for nearly integrable systems. This work studies the suitability of linear multistep methods for this kind of problems. It turns out that the symmetry of the method is essential for good conservation properties, and the more general class of partitioned linear multistep methods permits to obtain more favorable long-term stability of the integration. Insight into the long-time behavior is obtained by a backward error analysis, where the underlying one-step method and also parasitic solution components are investigated. In this way one approaches a classification of problems, for which multistep methods are an interesting class of integrators when long-time integration is important. Numerical experiments confirm the theoretical findings.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We thank Gustaf Söderlind for drawing our attention to this part of Dahlquist’s thesis.

  2. 2.

    A linear multistep is called strictly stable, if ζ 1 = 1 is a simple zero of the ρ polynomial, and all other zeros have modulus strictly smaller than one.

  3. 3.

    The case σ q (ζ l ) = 0 needs special attention, see the end of Sect. 3.5 or [9] for the special case of second order differential equations.

References

  1. M.J. Ablowitz, J.F. Ladik, A nonlinear difference scheme and inverse scattering. Stud. Appl. Math. 55, 213–229 (1976)

    MathSciNet  MATH  Google Scholar 

  2. G. Akrivis, M. Crouzeix, C. Makridakis, Implicit-explicit multistep methods for quasilinear parabolic equations. Numer. Math. 82(4), 521–541 (1999)

    Article  MathSciNet  Google Scholar 

  3. C. Arévalo, G. Söderlind, Convergence of multistep discretizations of DAEs. BIT 35(2), 143–168 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Chartier, E. Faou, A. Murua, An algebraic approach to invariant preserving integrators: The case of quadratic and Hamiltonian invariants. Numer. Math. 103, 575–590 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations. Transactions of the Royal Institute of Technology, vol. 130, Stockholm (1959)

    Google Scholar 

  6. K. Feng, Collected Works II (National Defense Industry Press, Beijing, 1995)

    Google Scholar 

  7. E. Hairer, Backward error analysis for multistep methods. Numer. Math. 84, 199–232 (1999)

    Article  MathSciNet  Google Scholar 

  8. E. Hairer, C. Lubich, Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2001)

    Article  MathSciNet  Google Scholar 

  9. E. Hairer, C. Lubich, Symmetric multistep methods over long times. Numer. Math. 97, 699–723 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Hairer, C. Schober, Corrigendum to: Symplectic integrators for the Ablowitz-Ladik discrete nonlinear Schrödinger equation [Phys. Lett. A 259 (1999), 140–151]. Phys. Lett. A 272(5–6), 421–422 (2000)

    Google Scholar 

  11. E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. Springer Series in Computational Mathematics, vol. 8, 2nd edn. (Springer, Berlin, 1993)

    Google Scholar 

  12. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31, 2nd edn. (Springer, Berlin, 2006)

    Google Scholar 

  13. E. Hairer, R. McLachlan, R. Skeel, On energy conservation of the simplified Takahashi–Imada method. ESAIM Math. Model. Numer. Anal. 43(4), 631–644 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. U. Kirchgraber, Multi-step methods are essentially one-step methods. Numer. Math. 48, 85–90 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)

    Article  Google Scholar 

  16. C.M. Schober, Symplectic integrators for the Ablowitz-Ladik discrete nonlinear Schrödinger equation. Phys. Lett. A 259, 140–151 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Y.-F. Tang, The symplecticity of multi-step methods. Comput. Math. Appl. 25, 83–90 (1993)

    Article  MATH  Google Scholar 

  18. D.S. Vlachos, T.E. Simos, Partitioned linear multistep method for long term integration of the N-body problem. Appl. Numer. Anal. Comput. Math. 1(3), 540–546 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to Luca Dieci and Nicola Guglielmi, organizers of the CIME-EMS summer school on “Current challenges in stability issues for numerical differential equations”, for including our research in their program. We acknowledge financial support from the Fonds National Suisse, Project No. 200020-126638 and 200021-129485.

Author information

Authors and Affiliations

  1. Section de Mathématiques, Université de Genève, Genève, Switzerland

    Paola Console & Ernst Hairer

Authors
  1. Paola Console
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ernst Hairer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paola Console .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Console, P., Hairer, E. (2014). Long-Term Stability of Symmetric Partitioned Linear Multistep Methods. In: Current Challenges in Stability Issues for Numerical Differential Equations. Lecture Notes in Mathematics(), vol 2082. Springer, Cham. https://doi.org/10.1007/978-3-319-01300-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation