BIT Numerical Mathematics

, Volume 47, Issue 2, pp 351–378 | Cite as


  • A. IserlesEmail author
  • G.R.W. Quispel
  • P.S.P. Tse


Volume preservation is one of the qualitative characteristics common to many dynamical systems. However, it has been proved by Kang and Shang that e.g. Runge–Kutta (RK) methods can not preserve volume for all linear source-free ODEs (let alone nonlinear ODEs). On the other hand, certain so-called Exponential Runge–Kutta (ERK) methods do preserve volume for all linear source-free ODEs. Do such ERK methods perhaps also preserve volume for all nonlinear ODEs? Here we prove that the answer to this question is negative; B-series methods (which include RK, ERK and several more classes of methods) cannot preserve volume for all source-free ODEs. The proof is presented via the theory of K-loops, which is an extension of the theory of classical rooted trees.

Key words

geometric integration volume preservation B-series methods modified equations 


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© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUnited Kingdom
  2. 2.Centre of Excellence for Mathematics and Statistics of Complex SystemsLa Trobe UniversityMelbourneAustralia
  3. 3.Department of MathematicsLa Trobe UniversityMelbourneAustralia

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