An optimum time-stepping scheme for N-body simulations

We present a new time-stepping criterion for N-body simulations that is based on the true dynamical time of a particle. This allows us to follow the orbits of particles correctly in all environments since it has better adaptivity than previous time-stepping criteria used in N-body simulations. Furthermore, it requires far fewer force evaluations in low-density regions of the simulation and has no dependence on artificial parameters such as, for example, the softening length. This can be orders of magnitude faster than conventional ad hoc methods that employ combinations of acceleration and softening and is ideally suited for hard problems, such as obtaining the correct dynamics in the very central regions of dark matter haloes. We also derive an eccentricity correction for a general leapfrog integration scheme that can follow gravitational scattering events for orbits with eccentricity e -> 1 with high precision. These new approaches allow us to study a range of problems in collisionless and collisional dynamics from few body problems to cosmological structure formation. We present tests of the time-stepping scheme in N-body simulations of two-body orbits with eccentricity e -> 1 (elliptic and hyperbolic), equilibrium haloes and a hierarchical cosmological structure formation run.