The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems

Abstract

The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems is an open and important problem in both numerical analysis and computing in science and engineering, as explicit integrators are usually more efficient than the implicit integrators of the same order of accuracy. Up to now, all responses to this problem are negative. That is, there exist explicit symplectic integrators only for some special nonseparable Hamiltonian systems, whereas the universal design involving explicit symplectic integrators for general nonseparable Hamiltonian systems has not yet been studied sufficiently. In this paper, we present a constructive proof for the existence of explicit symplectic integrators for general nonseparable Hamiltonian systems via finding explicit symplectic mappings under which the special submanifold of the extended phase space is invariant. It turns out that the proposed explicit integrators are symplectic in both the extended phase space and the original phase space. Moreover, on the basis of the global modified Hamiltonians of the proposed integrators, the backward error analysis is made via a parameter relaxation and restriction technique to show the linear growth of global errors and the near-preservation of first integrals. In particular, the effective estimated time interval is nearly the same as classical implicit symplectic integrators when applied to (near-) integrable Hamiltonian systems. Numerical experiments with a completely integrable nonseparable Hamiltonian and a nonintegrable nonseparable Hamiltonian illustrate the good long-term behavior and high efficiency of the explicit symplectic integrators proposed and analyzed in this paper.