BIT Numerical Mathematics

, Volume 55, Issue 4, pp 927–948

# Order conditions for G-symplectic methods

• John C. Butcher
Article

## Abstract

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.

## Keywords

G-symplectic methods Order conditions Conformability Trees Rooted trees

65L05 65L06

## Notes

### Acknowledgments

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

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