  In galactic nuclei, the gravitational potential is dominated by the central
supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits
are distorted by gravitational forces from other stars, leading to long-term
orbital relaxation. The direct numerical study of these processes is
challenging because the fast orbital motion imposed by the central black hole
requires very small timesteps. An alternative approach, pioneered by Gauss, is
to use the secular approximation of smearing out the $N$ stars over their
Keplerian orbits, using $K$ nodes along each orbit. In this study we propose
three novel improvements to this method. First, we re-formulate the
discretisation of the rates of change of the variables describing the orbital
states to ensure that all conservation laws are exactly satisfied. Second, we
replace 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 averaged
dynamical system is equivalent to $2N$ interacting unit spin vectors and
provide two time integrators: a second-order symplectic scheme and a
fourth-order Lie-group Runge-Kutta method, both of which are straightforward to
generalize to higher order. These new simulations recover the diffusion
coefficients of stellar eccentricities obtained through analytical calculations
of the secular dynamics.
