SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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The order of G-symplectic methods

J. Butcher and G. Imran

Abstract
G-symplectic general linear methods with zero parasitism growth factors are able to preserve quadratic invariants and symplectic structure in a similar way to canonical Runge-Kutta methods. To obtain order $p$ for a method $(A,U,B,V)$, it is necessary that $\xi$ exists such that $ (E\xi)(t) = B(D\eta)(t) + V \xi(t), $ for all $t$ such that $|t|\le p$, where $ \eta = A \eta D + U\xi. $ If the method is G-symplectic the order conditions are interrelated and can be reduced to a smaller set. The special case of order 4 methods with $V=\mbox{diag}(1,-1)$ will be considered in detail. Although there are only two order conditions, there are additional constraints on a possible starting method, represented by $\xi$. This should not be regarded as a serious handicap because, in a constant step size implementation, a complicated starting method does not represent a computational overhead.

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