  Self-interacting dark matter (SIDM) models have received great attention over
the past decade as solutions to the small-scale puzzles of astrophysics. Though
there are different implementations of dark matter (DM) self-interactions in
N-body codes of structure formation, there has not been a systematic study to
compare the predictions of these different implementations. We investigate the
implementation of dark matter self-interactions in two simulation codes: Gizmo
and Arepo. We begin with identical initial conditions for an isolated $10^{10}$
M$_\odot$ dark matter halo and investigate the evolution of the density and
velocity dispersion profiles in Gizmo and Arepo for SIDM cross-section over
mass of 1, 5, and 50 $\rm cm^2 g^{-1}$. Our tests are restricted to the core
expansion phase where the core density decreases and core radius increases with
time. We find better than 30% agreement between the codes for the density
profile in this phase of evolution, with the agreement improving at higher
resolution. We find that varying code-specific SIDM parameters changes the
central halo density by less than 10% outside of the convergence radius. We
argue that SIDM core formation is robust across the two different schemes and
conclude that these codes can reliably differentiate between cross-sections of
1, 5, and 50 $\rm cm^2 g^{-1}$ but finer distinctions would require further
investigation.
