Comparison of Runge-Kutta Algorithms and Symplectic Algorithms

The classical fourth-order Runge-Katla integrator and the third-order force gradient symplectic integrator are used to solve the two-dimensional H´enon-Heiles system respectively. Numerical results, including the relative energy error, Poincare section, the largest Lyapunov exponent and Fast Lyapunov Indicator, are compared in detail. It is found that the Runge-Katla algorithm does not conserve the energy of the system, but the symplectic one does. On the other hand, the former method gives some spurious descriptions of the dynamics, while the latter one does not.