Comparison between Symplectic Integrators and Clean Numerical Simulation for Chaotic Hamiltonian Systems

In this paper we compare the reliability of numerical simulations given by the classical symplectic integrator (SI) and the clean numerical simulation (CNS) for chaotic Hamiltonian systems. The chaotic Hénon-Heiles system and the famous three-body problem are used as examples for comparison. It is found that the numerical simulations given by the symplectic integrator indeed preserves the conservation of the total energy of system quite well. However, their orbits quickly depart away from each other. Thus, the SI can not give a reliable long-term evolution of orbits for these chaotic Hamiltonian systems. Fortunately, the CNS can give the convergent, reliable long-term evolution of solution trajectory with rather small deviations from the total energy. All of these suggest that the CNS could provide us a better and more reliable way than the SI to investigate chaotic Hamiltonian systems, from the microscopic quantum chaos to the macroscopic solar system.