Explicit canonical methods for Hamiltonian systems

Abstract

We consider canonical partitioned Runge-Kutta methods for separable Hamiltonians H = T ( p ) + V ( q ) H = T(p) + V(q) and canonical Runge-Kutta-Nyström methods for Hamiltonians of the form H = 1 2 p T M − 1 p + V ( q ) H = \frac {1}{2}{p^{\text {T}}}{M^{ - 1}}p + V(q) with M a diagonal matrix. We show that for explicit methods there is great simplification in their structure. Canonical methods of orders one through four are constructed. Numerical experiments indicate the suitability of canonical numerical schemes for long-time integrations.