BIT Numerical Mathematics

, Volume 48, Issue 2, pp 231–243 | Cite as

Achieving Brouwer’s law with implicit Runge–Kutta methods



In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article studies the influence of round-off on the conservation of first integrals such as the total energy in Hamiltonian systems. For implicit Runge–Kutta methods, a standard implementation shows an unexpected propagation. We propose a modification that reduces the effect of round-off and shows a qualitative and quantitative improvement for an accurate integration over long times.

Key words

round-off error probabilistic error propagation implicit Runge–Kutta methods long-time integration efficient implementation 


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  1. 1.
    D. Brouwer, On the accumulation of errors in numerical integration, Astron. J., 46 (1937), pp. 149–153.CrossRefGoogle Scholar
  2. 2.
    K. R. Grazier, W. I. Newman, J. M. Hyman, P. W. Sharp, and D. J. Goldstein, Achieving Brouwer’s law with high-order Störmer multistep methods, ANZIAM J., 46 (2004/05), pp. C786–C804.Google Scholar
  3. 3.
    E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn., Springer Ser. Comput. Math., vol. 31, Springer, Berlin, 2006.Google Scholar
  4. 4.
    P. Henrici, The propagation of round-off error in the numerical solution of initial value problems involving ordinary differential equations of the second order, in Symposium on the Numerical Treatment of Ordinary Differential Equations, Integral and Integro-Differential Equations (Rome 1960), Birkhäuser, Basel, 1960, pp. 275–291.Google Scholar
  5. 5.
    P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons Inc., New York, 1962.MATHGoogle Scholar
  6. 6.
    J. Laskar, P. Robutel, F. Joutel, M. Gastineau, A. C. M. Correia, and B. Levrard, A long-term numerical solution for the insolation quantities of the earth, Astron. Astrophys., 428 (2004), pp. 261–285.CrossRefGoogle Scholar
  7. 7.
    J.-M. Petit, Symplectic integrators: rotations and roundoff errors, Celest. Mech. Dyn. Astron., 70(1) (1998), pp. 1–21.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G. D. Quinlan, Round-off error in long-term orbital integrations using multistep methods, Celest. Mech. Dyn. Astron., 58(4) (1994), pp. 339–351.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  1. 1.Section de MathématiquesUniv. de GenèveGenève 4Switzerland
  2. 2.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand

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