Time-symmetric integration in astrophysics

Hernández, David M.; Bertschinger, Edmund

doi:10.1093/mnras/sty184

Cited by 22 publications

(23 citation statements)

References 24 publications

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“…The energy error of WHc is O( h 4 + 2 h 2 ). Applying correctors in this form has uses such as converting a reversible method to a symplectic one (Hairer et al 2006;Hernandez & Bertschinger 2018). LR has the same order energy error as WHc.…”

Section: Methodsmentioning

confidence: 99%

Are long-term N-body simulations reliable?

Hernández

Hadden

Makino

2020

Monthly Notices of the Royal Astronomical Society

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N-body integrations are used to model a wide range of astrophysical dynamics, but they suffer from errors which make their orbits diverge exponentially in time from the correct orbits. Over long time-scales, their reliability needs to be established. We address this reliability by running a three-body planetary system over about 200 e-folding times. Using nearby initial conditions, we can construct statistics of the long-term phase-space structure and compare to rough estimates of resonant widths of the system. Our statistics are approximately consistent for a wide range of numerical methods, including a Runge-Kutta method, Wisdom-Holman method, symplectic corrector methods, and a method by Laskar & Robutel. "Improving" an integrator did not affect the phase space accuracy, but simply increasing the number of initial conditions did.

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Section: Methodsmentioning

confidence: 99%

Are long-term N-body simulations reliable?

Hernández

Hadden

Makino

2020

Monthly Notices of the Royal Astronomical Society

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“…Although we have written the equations of motion for A and B, the integrators of (3) can be described by sets of ODEs called modified differential equations (MDEs) (Hairer et al 2006;Hernandez & Bertschinger 2018). We derive these for the firstorder integrator,…”

Section: Modified Differential Equationsmentioning

confidence: 99%

“…If an integrator is not Hamiltonian, we can still quantify its smoothness through its modified differential equations (e.g. Hairer et al 2006;Hernandez & Bertschinger 2018). Hernandez (2019) has explored the question of integrator smoothness for regular orbits.…”

Section: Introductionmentioning

confidence: 99%

“…According to the modified differential equations (MDEs) (e.g. Hernandez & Bertschinger 2018;Hairer et al 2006), we predict Hamiltonian smoothness should have a significant impact on the error of integrators. To test these ideas, we consider a restricted three-body problem in which the test particle executes a chaotic orbit in which it can visit both masses.…”

Section: Introductionmentioning

confidence: 99%

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Improving the accuracy of simulated chaotic N-body orbits using smoothness

Hernández

2019

Monthly Notices of the Royal Astronomical Society

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Symplectic integrators are a foundation to the study of dynamical N-body phenomena, at scales ranging from from planetary to cosmological. These integrators preserve the Poincaré invariants of Hamiltonian dynamics. The N-body Hamiltonian has another, perhaps overlooked, symmetry: it is smooth, or, in other words, it has infinite differentiability class order (DCO) for particle separations greater than 0. Popular symplectic integrators, such as hybrid methods or block adaptive stepping methods do not come from smooth Hamiltonians and it is perhaps unclear whether they should. We investigate the importance of this symmetry by considering hybrid integrators, whose DCO can be tuned easily. Hybrid methods are smooth, except at a finite number of phase space points. We study chaotic planetary orbits in a test considered by Wisdom. We find that increasing smoothness, at negligible extra computational cost in particular tests, improves the Jacobi constant error of the orbits by about 5 orders of magnitude in long-term simulations. The results from this work suggest that smoothness of the N-body Hamiltonian is a property worth preserving in simulations.

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“…See alsoHernandez & Bertschinger (2018) for error analysis of one-step methods from the perspective of their differential equations.…”

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confidence: 99%

REBOUNDx: a library for adding conservative and dissipative forces to otherwise symplectic N-body integrations

Tamayo

Rein

Shi

et al. 2019

Monthly Notices of the Royal Astronomical Society

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Symplectic methods, in particular the Wisdom-Holman map, have revolutionized our ability to model the long-term, conservative dynamics of planetary systems. However, many astrophysically important effects are dissipative. The consequences of incorporating such forces into otherwise symplectic schemes is not always clear. We show that moving to a general framework of non-commutative operators (dissipative or not) clarifies many of these questions, and that several important properties of symplectic schemes carry over to the general case. In particular, we show that explicit splitting schemes generically exploit symmetries in the applied external forces which often strongly suppress integration errors. Furthermore, we demonstrate that so-called 'symplectic correctors' (which reduce energy errors by orders of magnitude at fixed computational cost) apply equally well to weakly dissipative systems and can thus be more generally thought of as 'weak splitting correctors.' Finally, we show that previously advocated approaches of incorporating additional forces into symplectic methods work well for dissipative forces, but give qualitatively wrong answers for conservative but velocity-dependent forces like post-Newtonian corrections. We release REBOUNDx, an opensource C library for incorporating additional effects into REBOUND N-body integrations, together with a convenient PYTHON wrapper. All effects are machine-independent and we provide a binary format that interfaces with the SimulationArchive class in REBOUND to enable the sharing and reproducibility of results. Users can add effects from a list of pre-implemented astrophysical forces, or contribute new ones.

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Time-symmetric integration in astrophysics

Cited by 22 publications

References 24 publications

Are long-term N-body simulations reliable?

Are long-term N-body simulations reliable?

Improving the accuracy of simulated chaotic N-body orbits using smoothness

REBOUNDx: a library for adding conservative and dissipative forces to otherwise symplectic N-body integrations

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