Symplectic integrators in the shearing sheet

ABSTRACT

The shearing sheet is a model dynamical system that is used to study the small-scale dynamics of astrophysical discs. 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 the integrators are exact when there are no small-scale forces such as mutual gravitational forces between disc 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 discs in which the typical interparticle separation is large compared to the particles’ Hill radii (e.g., planetary rings), and the second one (‘SEKI’) is designed for discs 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).