A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes

Abstract

In recent publications, the construction of explicit symplectic integrators for Schwarzschild and Kerr type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various choices. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that the three-part splitting method is the best one of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge-Kutta and Runge-Kutta-Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The former algorithms need small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge-Kutta and Runge-Kutta-Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracies during long-term integrations due to roundoff errors. The idea finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including the Kerr type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge-Kutta and Runge-Kutta-Nyström methods are worth recommending.

arXiv comment10 pages, 5 figures. accepted for publication in ApJ

References in corpus (30)

• High order symplectic integrators for perturbed Hamiltonian systems
• Circular orbits and related quasi-harmonic oscillatory motion of charged particles around weakly magnetized rotating black holes
• Acceleration of charged particles due to chaotic scattering in the combined black hole gravitational field and asymptotically uniform magnetic field
• Weakly magnetized black holes as particle accelerators
• Transition from Regular to Chaotic Circulation in Magnetized Coronae near Compact Objects
• Explicit symplectic approximation of nonseparable Hamiltonians: algorithm and long time performance
• New families of symplectic splitting methods for numerical integration in dynamical astronomy
• Electrically charged matter in rigid rotation around magnetized black hole
• Chaotic motion of Charged Particles in an Electromagnetic Field Surrounding a Rotating Black Hole
• Pseudo-High-Order Symplectic Integrators
• Determination of chaotic behaviour in time series generated by charged particle motion around magnetized Schwarzschild black holes
• Explicit methods in extended phase space for inseparable Hamiltonian problems
• Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes
• Construction of explicit symplectic integrators in general relativity. II. Reissner-Nordstrom black holes
• On post-Newtonian orbits and the Galactic-center stars
• Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes
• Explicit high-order symplectic integrators for charged particles in general electromagnetic fields
• Inducing chaos by breaking axial symmetry in a black hole magnetosphere
• Construction of explicit symplectic integrators in general relativity. III. Reissner-Nordstrom-(anti)-de Sitter black holes
• Symplectic Integration of Post-Newtonian Equations of Motion with Spin
• Evolution and Distribution of Magnetic Fields from AGNs in Galaxy Clusters II. The Effects of Cluster Size and Dynamical State
• Near-horizon structure of escape zones of electrically charged particles around weakly magnetized rotating black hole
• The Midpoint Rule as a Variational--Symplectic Integrator. I. Hamiltonian Systems
• A Symmetric Integrator for non-integrable Hamiltonian Relativistic Systems
• Applying explicit symplectic integrator to study chaos of charged particles around magnetized Kerr black hole
• Gauss collocation methods for efficient structure preserving integration of post-Newtonian equations of motion
• Extended phase-space symplectic-like integrators for coherent post-Newtonian Euler-Lagrange equations
• Dynamics of charged particles moving around Kerr black hole with inductive charge and external magnetic field
• Charged particle motions near non-Schwarzschild black holes with external magnetic fields in modified theories of gravity
• GRIT: a package for structure-preserving simulations of gravitationally interacting rigid-bodies

Cited by in corpus (16)

• Structure-preserving numerical simulations of test particle dynamics around slowly rotating neutron stars within Hartle-Thorne approach
• Explicit symplectic methods in black hole spacetimes
• Electromagnetic field and chaotic charged-particle motion around hairy black holes in Horndeski gravity
• Chaos in a Magnetized Modified Gravity Schwarzschild Spacetime
• Equivalence between two charged black holes in dynamics of orbits outside the event horizons
• Semiexplicit Symplectic Integrators for Non-separable Hamiltonian Systems
• Impact of electric charges on chaos in magnetized $Reissner-Nordstr\ddot{o}m$ spacetimes
• Preservation of Quadratic Invariants by Semiexplicit Symplectic Integrators for Non-separable Hamiltonian Systems
• Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems
• Mutual Information for particle pair and its application to diagnose Chaos in Curved Spacetime
• Screen chaotic motion by Shannon entropy in curved spacetimes
• Study of chaos in rotating galaxies using extended force-gradient symplectic methods
• Measurement of the central Galactic black hole by extremely large mass-ratio inspirals
• Contrasting the Implicit Method in Incoherent Lagrangian and the Correction Map Method in Hamiltonian
• Dissipated Correction Map Method with Trapezoidal Rule for the Simulations of Gravitational Waves from Spinning Compact Binary
• Chaotic dynamics of charged particles near weakly magnetized black holes in Einstein-ModMax Theory