Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems

摘要

In this paper we compare the reliability of numerical simulations given bythe classical symplectic integrator (SI) and the clean numerical simulation(CNS) for chaotic Hamiltonian systems. The chaotic H'{e}non-Heiles system andthe famous three-body problem are used as examples for comparison. It is foundthat the numerical simulations given by the symplectic integrator indeedpreserves 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 areliable long-term evolution of orbits for these chaotic Hamiltonian systems.Fortunately, the CNS can give the convergent, reliable long-term evolution ofsolution trajectory with rather small deviations from the total energy. All ofthese suggest that the CNS could provide us a better and more reliable way thanthe SI to investigate chaotic Hamiltonian systems, from the microscopic quantumchaos to the macroscopic solar system.