High-Order Symplectic Runge–Kutta–Nyström Methods

Abstract

A numerical method for ordinary differential equations is called symplectic if, when applied to Hamiltonian problems, it preserves the symplectic structure in phase space, thus reproducing the main qualitative property of solutions of Hamiltonian systems. The authors construct and test symplectic, explicit Runge–Kutta–Nyström (RKN) methods of order 8. The outcome of the investigation is that existing high-order, symplectic RKN formulae require so many evaluations per step that they are much less efficient than conventional eighth-order nonsymplectic, variable-step-size integrators even for low accuracy. However, symplectic integration is of use in the study of qualitative features of the systems being integrated.