Sympletic Runge--Kutta Shemes I: Order Conditions

Abstract

Much recent work has indicated that considerable benefit arises from the use of symplectic algorithms when numerically integrating Hamiltonian systems of differential equations. Runge--Kutta schemes are symplectic subject to a simple algebraic condition. Starting with Butcher's formalism it is shown that there exists a more natural basis for the set of necessary and sufficient order conditions for these methods, involving only s(s + 1)/2 free parameters for a symplectic s-stage scheme. A graph theoretical process for determining the new order conditions is outlined. Furthermore, it is shown that any rooted tree arising from the same free tree enforces the same algebraic constraint on the parametrized coefficients. When coupled with the standard simplifying assumptions for implicit schemes the number of order conditions may be further reduced. In the new framework a simple symmetry of the parameter matrix yields (not necessarily symplectic) self-adjoint methods. In this case the order conditions associated with even trees become redundant.