IAS15: A fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits (1409.4779v2)

Abstract: We present IAS15, a 15th-order integrator to simulate gravitational dynamics. The integrator is based on a Gau\ss-Radau quadrature and can handle conservative as well as non-conservative forces. We develop a step-size control that can automatically choose an optimal timestep. The algorithm can handle close encounters and high-eccentricity orbits. The systematic errors are kept well below machine precision and long-term orbit integrations over $10^9$ orbits show that IAS15 is optimal in the sense that it follows Brouwer's law, i.e. the energy error behaves like a random walk. Our tests show that IAS15 is superior to a mixed-variable symplectic integrator (MVS) and other popular integrators, including high-order ones, in both speed and accuracy. In fact, IAS15 preserves the symplecticity of Hamiltonian systems better than the commonly-used nominally symplectic integrators to which we compared it. We provide an open-source implementation of IAS15. The package comes with several easy-to-extend examples involving resonant planetary systems, Kozai-Lidov cycles, close encounters, radiation pressure, quadrupole moment, and generic damping functions that can, among other things, be used to simulate planet-disc interactions. Other non-conservative forces can be added easily.

• The paper introduces a 15th-order integrator based on Gau quadrature that adaptively controls timesteps to maintain accuracy at machine precision over extensive orbital simulations.
• The paper demonstrates that IAS15 outperforms traditional methods like Wisdom-Holman and Bulirsch-Stoer integrators in both speed and energy error control over billions of orbits.
• The paper shows that the integrator effectively handles both conservative and non-conservative forces, making it an invaluable tool for long-term gravitational dynamics studies.

Summary of "A Fast, Adaptive, High-Order Integrator for Gravitational Dynamics, Accurate to Machine Precision Over a Billion Orbits"

The paper by Hanno Rein and David Spiegel introduces a 15th-order integrator designed for simulating gravitational dynamics with high precision, aptly named the Implicit integrator with Adaptive timeStepping (IAS). This integrator is built on a foundation of Gau quadrature and aims to address limitations in conventional symplectic integrators. It offers a robust solution for both conservative and non-conservative forces, with the capability to adaptively control timesteps based on error estimates. Notably, the authors have demonstrated its efficacy over a billion orbital periods with systematic errors kept beneath machine precision.

Key Features and Results

• Integrator Design: The paper describes an integrator based on a Gau quadrature scheme, which is shown to handle both conservative and non-conservative forces effectively. Its high-order nature is advantageous when dealing with orbital mechanics, allowing for accuracy well below machine precision across extensive simulations. The authors underline that the integrator satisfies Brouwer's law, with energy error behaving like a random walk.
• Comparison with Existing Integrators: The authors conducted a series of tests comparing their integrator against other popular methods, including the Wisdom-Holman (WH) mixed-variable symplectic (MVS) integrator and Bulirsch-Stoer (BS) integrators. The results persistently favored the new integrator in both speed and accuracy. Specifically, the IAS demonstrated superior symplecticity preservation and practical usability over massively extended simulation periods.
• Adaptive Time-Stepping: A salient feature highlighted is the adaptive step-size control system that chooses optimal time steps without needing prior system knowledge. This is particularly important for long-term stability in simulations involving planetary resonances, Kozai-Lidov cycles, and close encounters.
• Error Analysis and Scaling: Intensive error assessments reveal that the IAS follows Brouwer's law accurately, where error growth adheres to expected random walk scaling. These findings are backed by examinations of the floating-point precision impacts on error growth (quantified as $E_{\rm floor}$, $E_{\rm rand}$, among others), illuminating the intricate balance between systematic integrator error and machine-specific errors.

Implications and Future Directions

The development of the IAS integrator represents a significant step in celestial mechanics simulation, aiming to deliver unmatched accuracy while mitigating machine precision constraints. Practically, this makes it an invaluable tool for the astrophysics community, enabling precise studies of long-term solar system evolution and other complex dynamical systems.

Theoretically, the results suggest possible extensions into simulations involving higher-order non-conservative forces. With optimization, such integrators could transform computational approaches in evolving areas of astrophysics and cosmology.

Future work could involve leveraging the framework of this integrator to explore integrations in systems with additional factors, like significant damping forces or relativistic corrections. Furthermore, exploring the integration of this system on architectures with varying precision levels, such as potential implementations on GPU-based platforms or extended precision computational setups, could offer enhanced performance benefits.

The paper's contribution lies in its meticulous construction of a highly adaptive, high-order integrator, which responds dynamically to computational demands late into simulations — a characteristic that marks a forward-thinking approach to gravitational simulations. The released open-source implementation further supports adoption and experimentation by the research community.