Time-symmetric integration in astrophysics

Abstract

Calculating the long term solution of ordinary differential equations, such as those of the $N$-body 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.

15 pages, 10 figures, 3, tables. Matches accepted MNRAS version

References in corpus (8)

• The cosmological simulation code GADGET-2
• N-body Integrators with Individual Time Steps from Hierarchical Splitting
• Time-Symmetrized Kustaanheimo-Stiefel Regularization
• A Time-Symmetric Block Time-Step Algorithm for N-Body Simulations
• Symplectic fourth-order maps for the collisional N-body problem
• Symplectic integration for the collisional gravitational $N$-body problem
• Fast and reliable symplectic integration for planetary system $N$-body problems
• Towards time symmetric N-body integration

Cited by in corpus (15)

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• TRACE: a code for Time-Reversible Astrophysical Close Encounters
• Structure-preserving integrators for dissipative systems based on reversible-irreversible splitting
• Hypersolvers: Toward Fast Continuous-Depth Models
• Are long-term $N$-body simulations reliable?
• A differentiable N-body code for transit timing and dynamical modeling. I. Algorithm and derivatives
• Stepsize errors in the $N$-body problem: discerning Mercury's true possible long-term orbits
• Should $N$-body integrators be symplectic everywhere in phase space?
• Switching integrators reversibly in the astrophysical $N$-body problem
• Errors, chaos and the collisionless limit
• EnckeHH: an integrator for gravitational dynamics with a dominant mass that achieves optimal error behaviour
• Improving the accuracy of simulated chaotic $N$-body orbits using smoothness
• Understanding the evolution of Atira-class asteroid 2019 AQ3, a major step towards the future discovery of the Vatira population