A momentum conserving $N$-body scheme with individual timesteps

$N$-body simulations study the dynamics of $N$ particles under the influence of mutual long-distant forces such as gravity. In practice, $N$-body codes will violate Newton's third law if they use either an approximate Poisson solver or individual timesteps. In this study, we construct a novel $N$-body scheme by combining a fast multipole method (FMM) based Poisson solver and a time integrator using a hierarchical Hamiltonian splitting (HHS) technique. We test our implementation for collision-less systems using several problems in galactic dynamics. As a result of the momentum conserving nature of these two key components, the new $N$-body scheme is also momentum conserving. Moreover, we can fully utilize the $\mathcal O(\textit N)$ complexity of FMM with the integrator. With the restored force symmetry, we can improve both angular momentum conservation and energy conservation substantially. The new scheme will be suitable for many applications in galactic dynamics and structure formation. Our implementation, in the code Taichi, is publicly available at https://bitbucket.org/qirong_zhu/taichi_public/.