  Analysis of large galaxy surveys requires confidence in the robustness of
numerical simulation methods. The simulations are used to construct mock galaxy
catalogs to validate data analysis pipelines and identify potential
systematics. We compare three $N$-body simulation codes, ABACUS, GADGET, and
SWIFT, to investigate the regimes in which their results agree. We run $N$-body
simulations at three different mass resolutions, $6.25\times10^{8}$,
$2.11\times10^{9}$, and $5.00\times10^{9}~h^{-1}$M$_{\odot}$, matching phases
to reduce the noise within the comparisons. We find systematic errors in the
halo clustering between different codes are smaller than the DESI statistical
error for $s > 20\, h^{-1}$Mpc in the correlation function in redshift space.
Through the resolution comparison we find that simulations run with a mass
resolution of $2.1\times10^{9}~h^{-1}$M$_{\odot}$ are sufficiently converged
for systematic effects in the halo clustering to be smaller than the DESI
statistical error at scales larger than $20 \, h^{-1}$Mpc. These findings show
that the simulations are robust for extracting cosmological information from
large scales which is the key goal of the DESI survey. Comparing matter power
spectra, we find the codes agree to within 1% for $k \leq 10~h$Mpc$^{-1}$. We
also run a comparison of three initial condition generation codes and find good
agreement. In addition, we include a quasi-$N$-body code, FastPM, since we plan
use it for certain DESI analyses. The impact of the halo definition and
galaxy-halo relation will be presented in a follow up study.
