BIT Numerical Mathematics

, Volume 55, Issue 4, pp 927–948 | Cite as

Order conditions for G-symplectic methods

  • John C. Butcher
  • Gulshad Imran


General linear methods for the solution of ordinary differential equations are both multivalue and multistage. The order conditions will be stated and analyzed using a B-series approach. However, imposing the G-symplectic structure modifies the nature of the order conditions considerably. For Runge–Kutta methods, rooted trees belonging to the same tree have equivalent order conditions; if the trees are superfluous, they are automatically satisfied and can be ignored. For G-symplectic methods, similar results apply but with a more general interpretation. In the multivalue case, starting conditions are a natural aspect of the meaning of order; unlike the Runge–Kutta case for which “effective order” or “processing” or “conjugacy” has to be seen as having an artificial meaning. It is shown that G-symplectic methods with order 4 can be constructed with relatively few stages, \(s=3\), and with only \(r=2\) inputs to a step.


G-symplectic methods Order conditions Conformability Trees Rooted trees 

Mathematics Subject Classification

65L05 65L06 



The authors acknowledge the support of the Marsden Fund. Furthermore, they thank the referees for their valuable comments.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand

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