Further reduction in the number of independent order conditions for symplectic, explicit partitioned Runge-Kutta and Runge-Kutta-Nyström methods (Q1902067)

Explicit partitioned symplectic Runge-Kutta methods for a separable Hamiltonian system \(H(p,q)=T(p)+V(q)\), can be analysed using bicolour rooted trees. Using this approach, the conditions for the method to be symplectic act as simplifying assumptions and allow many of the rooted trees to be eliminated. An alternative analysis by \textit{R. I. McLachlan} [SIAM J. Sci. Comput. 16, No. 1, 151-168 (1995; Zbl 0821.65048)], using free Lie algebra theory, where the methods are interpreted as composition methods, yields a smaller set of conditions for orders greater than 5. The main result of the present paper is a relation between the order conditions associated with various sets of four trees. This permits the elimination of further trees and, for orders as high as 10, brings the number of independent conditions down to the number found using Lie theory. In the important special case \(T(p)={1\over2}p^Tp\), Runge-Kutta-Nyström methods are available and these can be interpreted using the explicit partitioned Runge-Kutta method approach but with further trees removed because of the quadratic form of the kinetic energy \(T(p)\). The reduction in the set of order conditions, using the relations proved in this paper, brings the number of conditions down to 64 for order 9 and 103 for order 10. This compares favourably with the Lie theory results, which yield 65 conditions for order 9 and 105 for order 10.