  The ongoing merger of the Sagittarius (Sgr) dwarf galaxy with the Milky Way
is believed to strongly affect the dynamics of the Milky Way's disc. We present
a suite of 13 $N$-body simulations, with 500 million to 1 billion particles,
modelling the interaction between the Sagittarius dwarf galaxy (Sgr) and the
Galactic disc. To quantify the perturbation to the disc's structure and
dynamics in the simulation, we compute the number count asymmetry and the mean
vertical velocity in a solar-neighbourhood-like volume. We find that overall
the trends in the simulations match those seen in a simple one-dimensional
model of the interaction. We explore the effects of changing the mass model of
Sgr, the orbital kinematics of Sgr, and the mass of the Milky Way halo. We find
that none of the simulations match the observations of the vertical
perturbation using Gaia Data Release 2. In the simulation which is the most
similar, we find that the final mass of Sgr far exceeds the observed mass of
the Sgr remnant, the asymmetry wavelength is too large, and the shape of the
asymmetry doesn't match past $z \approx 0.7$ kpc. We therefore conclude that
our simulations support the conclusion that Sgr alone could not have caused the
observed perturbation to the solar neighbourhood.
