The $\texttt{Abacus}$ Cosmological $N$-body Code

We present $\texttt{Abacus}$, a fast and accurate cosmological $N$-body code based on a new method for calculating the gravitational potential from a static multipole mesh. The method analytically separates the near- and far-field forces, reducing the former to direct $1/r^2$ summation and the latter to a discrete convolution over multipoles. The method achieves 70 million particle updates per second per node of the Summit supercomputer, while maintaining a median fractional force error of $10^{-5}$. We express the simulation time step as an event-driven "pipeline", incorporating asynchronous events such as completion of co-processor work, Input/Output, and network communication. $\texttt{Abacus}$ has been used to produce the largest suite of $N$-body simulations to date, the $\texttt{AbacusSummit}$ suite of 60 trillion particles (Maksimova et al., 2021), incorporating on-the-fly halo finding. $\texttt{Abacus}$ enables the production of mock catalogs of the volume and resolution required by the coming generation of cosmological surveys.