The non-existence of symplectic multi-derivative Runge-Kutta methods (Q1343042)

The authors investigate the numerical solution of a Hamiltonian system, especially the property called``symplecticness'' of a numerical method which consists in the preservation of some differential 2-form. First \textit{F. M. Lasagni} [Integration methods for Hamiltonian differential equations. (Unpublished manuscript)] has studied this property of multi- derivative (\(q\)) Runge-Kutta methods. The main results of this paper are: 1) It is shown that an irreducible Runge-Kutta method can be symplectic only for \(q \leq 1\), i.e., for standard Runge-Kutta methods. 2) It is shown that in this case \((q\leq 1)\) the conditions of Lasagni for symplecticness are also necessary, so there are no symplectic multi- derivative Runge-Kutta methods.