BIT Numerical Mathematics

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

Achieving Brouwer’s law with implicit Runge–Kutta methods

  • E. Hairer
  • R. I. McLachlan
  • A. Razakarivony


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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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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