Construction of second-order six-dimensional Hamiltonian-conserving scheme

TL;DR

This paper introduces a second-order energy-conserving implicit numerical scheme for six-dimensional Hamiltonian systems, validated through simulations of astrophysical models, and explores chaos phenomena influenced by system parameters.

Contribution

A novel second-order energy-conserving implicit scheme for six-dimensional Hamiltonian systems is developed, improving accuracy over existing methods.

Findings

The new scheme achieves second-order accuracy in energy conservation.

Chaos in astrophysical models depends on parameters like mass, perturbation, and magnetic field.

The method is applicable to various six-dimensional Hamiltonian problems, including spacetimes with (3+1) splits.

Abstract

It is shown analytically that the energy-conserving implicit nonsymplectic scheme of Bacchini, Ripperda, Chen and Sironi provides a first-order accuracy to numerical solutions of a six-dimensional conservative Hamiltonian system. Because of this, a new second-order energy-conserving implicit scheme is proposed. Numerical simulations of Galactic model hosting a BL Lacertae object and magnetized rotating black hole background support these analytical results. The new method with appropriate time steps is used to explore the effects of varying the parameters on the presence of chaos in the two physical models. Chaos easily occurs in the Galactic model as the mass of the nucleus, the internal perturbation parameter, and the anisotropy of the potential of the elliptical galaxy increase. The dynamics of charged particles around the magnetized Kerr spacetime is easily chaotic for larger energies of the particles, smaller initial angular momenta of the particles, and stronger magnetic fields. The chaotic properties are not necessarily weakened when the black hole spin increases. The new method can be used for any six-dimensional Hamiltonian problems, including globally hyperbolic spacetimes with readily available (3+1) split coordinates.