Symplectic integrators in the shearing sheet

TL;DR

This paper introduces two new symplectic, time-reversible integrators for simulating particle dynamics in the shearing sheet model, significantly improving accuracy over traditional leapfrog methods especially in astrophysical disk simulations.

Contribution

The paper presents two novel integrators, SEI and SEKI, tailored for different disk conditions, enhancing simulation precision and efficiency in astrophysical research.

Findings

Integrators outperform existing methods by several orders of magnitude in accuracy.

SEI is optimal for large inter-particle separations, like planetary rings.

SEKI is designed for tightly bound particles, such as irregular satellites.

Abstract

The shearing sheet is a model dynamical system that is used to study the small-scale dynamics of astrophysical disks. Numerical simulations of particle trajectories in the shearing sheet usually employ the leapfrog integrator, but this integrator performs poorly because of velocity-dependent (Coriolis) forces. We describe two new integrators for this purpose; both are symplectic, time-reversible and second-order accurate, and can easily be generalized to higher orders. Moreover, both integrators are exact when there are no small-scale forces such as mutual gravitational forces between disk particles. In numerical experiments these integrators have errors that are often several orders of magnitude smaller than competing methods. The first of our new integrators (SEI) is well-suited for disks in which the typical inter-particle separation is large compared to the particles' Hill radii (e.g., planetary rings), and the second (SEKI) is designed for disks in which the particles are on bound orbits or the separation is smaller than the Hill radius (e.g., irregular satellites of the giant planets).