Energy-conserving integrator for conservative Hamiltonian systems with ten-dimensional phase space

TL;DR

This paper introduces an implicit energy-preserving integrator for ten-dimensional Hamiltonian systems, demonstrating excellent long-term energy conservation and effectiveness in analyzing chaos in complex dynamical systems.

Contribution

The paper presents a novel second-order, implicit, nonsymplectic energy-preserving integrator specifically designed for high-dimensional Hamiltonian systems, with demonstrated numerical advantages.

Findings

The integrator maintains energy conservation over long simulations.

Large time steps reduce computational cost and roundoff errors.

The method effectively detects chaos in complex dynamical systems.

Abstract

In this paper, an implicit nonsymplectic exact energy-preserving integrator is specifically designed for a ten-dimensional phase-space conservative Hamiltonian system with five degrees of freedom. It is based on a suitable discretization-averaging of the Hamiltonian gradient, with a second-order accuracy to numerical solutions. A one-dimensional disordered discrete nonlinear Schr\"{o}dinger equation and a post-Newtonian Hamiltonian system of spinning compact binaries are taken as our two examples. We demonstrate numerically that the proposed algorithm exhibits good long-term performance in the preservation of energy, if roundoff errors are neglected. This result is independent of time steps, initial orbital eccentricities, and regular and chaotic orbital dynamical behavior. In particular, the application of appropriately large time steps to the new algorithm is helpful in reducing time-consuming and roundoff errors. This new method, combined with fast Lyapunov indicators, is well suited related to chaos in the two example problems. It is found that chaos in the former system is mainly responsible for one of the parameters. In the latter problem, a combination of small initial separations and high initial eccentricities can easily induce chaos.