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

TL;DR

IAS15 is a 15th-order adaptive integrator for gravitational dynamics that achieves machine-precision accuracy over billions of orbits, outperforming existing methods in speed and precision, and is suitable for complex astrophysical simulations.

Contribution

We introduce IAS15, a high-order, adaptive integrator based on Gau extbackslash{}ss-Radau quadrature, capable of handling conservative and non-conservative forces with superior accuracy and efficiency.

Findings

Systematic errors are below machine precision.

Outperforms symplectic integrators in speed and accuracy.

Maintains better symplecticity than nominally symplectic methods.

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.