Numerische Mathematik

, Volume 97, Issue 4, pp 699–723 | Cite as

Symmetric multistep methods over long times

  • Ernst HairerEmail author
  • Christian Lubich


For computations of planetary motions with special linear multistep methods an excellent long-time behaviour is reported in the literature, without a theoretical explanation. Neither the total energy nor the angular momentum exhibit secular error terms. In this paper we completely explain this behaviour by studying the modified equation of these methods and by analyzing the remarkably stable propagation of parasitic solution components.


Total Energy Angular Momentum Error Term Theoretical Explanation Stable Propagation 
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  1. 1.
    Benettin, G., Giorgilli, A.: On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J. Statist. Phys. 74, 1117–1143 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cano, B., Sanz-Serna, J. M.: Error growth in the numerical integration of periodic orbits by multistep methods with application to reversible systems. IMA J. Numer. Anal. 18, 57–75 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. of the Royal Inst. of Techn., Stockholm, Sweden 130, 1959Google Scholar
  4. 4.
    Evans, N. W., Tremaine, S.: Linear multistep methods for integrating reversible differential equations. Astron. J. 118, 1888–1899 (1999)CrossRefGoogle Scholar
  5. 5.
    Fukushima, T.: Symmetric multistep methods revisited. In 30th Symposium on Celestial Mechanics, 1998, pp. 229–247Google Scholar
  6. 6.
    Fukushima, T.: Symmetric multistep methods revisited: II. Numerical experiments. In 173rd colloquium of the International Astronomical Union, 1999, pp. 309–314Google Scholar
  7. 7.
    Hairer, E.: Backward error analysis for multistep methods. Numer. Math. 84, 199–232 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Hairer, E., Hairer, M.: GniCodes – Matlab programs for geometric numerical integration. In: Frontiers in Numerical Analysis (Durham 2002), Springer, Berlin, 2003Google Scholar
  9. 9.
    Hairer, E., Leone, P.: Order barriers for symplectic multi-value methods. In: Numerical analysis 1997, Proc. of the 17th Dundee Biennial Conference 1997, D. F. Griffiths D. J. Higham & G. A. Watson eds. Pitman Research Notes in Mathematics Series. 380, 133–149 1998Google Scholar
  10. 10.
    Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics 31. Springer, Berlin, 2002Google Scholar
  12. 12.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numerica 12, 2003Google Scholar
  13. 13.
    Hairer, E., Nørsett, S. P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Springer Series in Computational Mathematics 8. Springer, Berlin, 2nd edition, 1993Google Scholar
  14. 14.
    Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. John Wiley & Sons Inc., New York, 1962Google Scholar
  15. 15.
    Lambert, J. D., Watson, I. A.: Symmetric multistep methods for periodic initial value problems. J. Inst. Maths. Applics. 18, 189–202 (1976)zbMATHGoogle Scholar
  16. 16.
    Moser, J.: Stable and random motions in dynamical systems. Annals of Mathematics Studies. 77, 1973Google Scholar
  17. 17.
    Quinlan, G. D., Tremaine, S.: Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)CrossRefGoogle Scholar
  18. 18.
    Störmer, C.: Sur les trajectoires des corpuscules électrisés. Arch. sci. phys. nat. Genève 24, 5–18, 113–158, 221–247 (1907)Google Scholar
  19. 19.
    Tang, Y.-F.: The symplecticity of multi-step methods. Computers Math. Applic. 25, 83–90 (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Dept. de MathématiquesUniv. de GenèveGenève 24Switzerland
  2. 2.Mathematisches InstitutUniv. TübingenTübingenGermany

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