Fast and reliable symplectic integration for planetary system N -body problems

Monthly Notices of the Royal Astronomical Society

Hadden

Makino

2020

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.

Section: Introductionmentioning

confidence: 99%

Are long-term N-body simulations reliable?

Monthly Notices of the Royal Astronomical Society

Hadden

Makino

2020

“…Depending on the equations, geometric properties include the Hamiltonian flow, time-reversibility, and quadratic and linear invariants in the phase space. In the last 30 years, astrophysics researchers have made geometric integration a standard in various fields of dynamics, including planets (Wisdom & Holman 1991;Chambers 1999;Duncan et al 1998;Hernandez 2016), stellar clusters (Kokubo et al 1998;Hut et al 1995;Hernandez & Bertschinger 2015;Dehnen & Hernandez 2017), or galaxy formation (Springel 2005).…”

Section: Introductionmentioning

confidence: 99%

Time-symmetric integration in astrophysics

Monthly Notices of the Royal Astronomical Society

Bertschinger

2018

Calculating the long term solution of ordinary differential equations, such as those of the Nbody problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally no analytic solution exists to these equations, researchers rely on numerical methods which are prone to various errors. In an effort to mitigate these errors, powerful symplectic integrators have been employed. But symplectic integrators can be severely limited because they are not compatible with adaptive stepping and thus they have difficulty accommodating changing time and length scales. A promising alternative is time-reversible integration, which can handle adaptive time stepping, but the errors due to time-reversible integration in astrophysics are less understood. The goal of this work is to study analytically and numerically the errors caused by time-reversible integration, with and without adaptive stepping. We derive the modified differential equations of these integrators to perform the error analysis. As an example, we consider the trapezoidal rule, a reversible non-symplectic integrator, and show it gives secular energy error increase for a pendulum problem and for a Hénon-Heiles orbit. We conclude that using reversible integration does not guarantee good energy conservation and that, when possible, use of symplectic integrators is favored. We also show that time-symmetry and time-reversibility are properties that are distinct for an integrator.

“…We tested this alternative splitting, but it did not perform better in the tests in this paper because solving the n = 1 problem is not the only consideration for a good integrator (Hernandez & Dehnen 2017, Figure 6, and error analysis).H is given by the BCH formula (e.g. Hairer et al 2006) as,…”

Section: Algorithm Descriptionmentioning

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%

See 1 more Smart Citation

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

Monthly Notices of the Royal Astronomical Society

2019

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.