Partitioned Runge-Kutta methods for separable Hamiltonian problems

Abstract

Separable Hamiltonian systems of differential equations have the form d p / d t = − ∂ H / ∂ q d{\mathbf {p}}/dt = - \partial H/\partial {\mathbf {q}} , d q / d t = ∂ H / ∂ p d{\mathbf {q}}/dt = \partial H/\partial {\mathbf {p}} , with a Hamiltonian function H that satisfies H = T ( p ) + V ( q ) H = T({\mathbf {p}}) + V({\mathbf {q}}) (T and V are respectively the kinetic and potential energies). We study the integration of these systems by means of partitioned Runge-Kutta methods, i.e., by means of methods where different Runge-Kutta tableaux are used for the p and q equations. We derive a sufficient and "almost" necessary condition for a partitioned Runge-Kutta method to be canonical, i.e., to conserve the symplectic structure of phase space, thereby reproducing the qualitative properties of the Hamiltonian dynamics. We show that the requirement of canonicity operates as a simplifying assumption for the study of the order conditions of the method.