Performance of variable step size methods for solving model separable hamiltonian systems

We compare adaptive step size methods which approximately conserve a separable Hamiltonian with variable step size methods designed to integrate accurately the associated Hamiltonian system of ordinary differential equations (ODEs). Particularly, we consider integrating the ODEs describing the planar three body problem. We show that, a second order variable step size Verlet method while approximately conserving the Hamiltonian is unable to reproduce the solution accurately or efficiently. We demonstrate the failure of this method on the Hénon-Heiles problem. Then, we turn to symplectic variable step size implicit Runge-Kutta (IRK) methods and compare them with variable step size implementations of Runge-Kutta Nystrِm (RKN) methods optimized for accuracy only. Our measure of accuracy is the codesʹ ability to conserve the Hamiltonian whilst computing a qualitatively correct solution. Also, we show that not solving the (nonlinear) IRK equations exactly can lead to nonconservation.