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In galactic nuclei, the gravitational potential is dominated by the centralsupermassive black hole, so stars follow quasi-Keplerian orbits. These orbitsare distorted by gravitational forces from other stars, leading to long-termorbital relaxation. The direct numerical study of these processes ischallenging because the fast orbital motion imposed by the central black holerequires very small timesteps. An alternative approach, pioneered by Gauss, isto use the secular approximation of smearing out the $N$ stars over theirKeplerian orbits, using $K$ nodes along each orbit. In this study we proposethree novel improvements to this method. First, we re-formulate thediscretisation of the rates of change of the variables describing the orbitalstates to ensure that all conservation laws are exactly satisfied. Second, wereplace the pairwise sum over nodes by a multipole expansion up to order$\ell_{\mathrm{max}}$, reducing the overall computational costs from$O(N^2K^2)$ to $O(NK\ell_{\mathrm{max}}^2)$. Finally, we show that the averageddynamical system is equivalent to $2N$ interacting unit spin vectors andprovide two time integrators: a second-order symplectic scheme and afourth-order Lie-group Runge-Kutta method, both of which are straightforward togeneralize to higher order. These new simulations recover the diffusioncoefficients of stellar eccentricities obtained through analytical calculationsof the secular dynamics.

Secular Dynamics around a Supermassive Black Hole via Multipole Expansion