SYMPLECTIC RUNGE-KUTTA SCHEMES .1. ORDER CONDITIONS

Much recent work has indicated that considerable benefit arises from t he use of symplectic algorithms when numerically integrating Hamiltoni an systems of differential equations. Runge-Kutta schemes are symplect ic subject to a simple algebraic condition. Starting with Butcher's fo rmalism it is shown that there exists a more natural basis for the set of necessary and sufficient order conditions for these methods, invol ving only s(s + 1)/2 free parameters for a symplectic s-stage scheme. A graph theoretical process for determining the new order conditions i s outlined. Furthermore, it is shown that any rooted tree arising from the same free tree enforces the same algebraic constraint on the para metrized coefficients. When coupled with the standard simplifying assu mptions for implicit schemes the number of order conditions may be fur ther reduced. In the new frame work a simple symmetry of the parameter matrix yields (not necessarily symplectic) self-adjoint methods. In t his case the order conditions associated with even trees become redund ant.