  We examine the effect of spatial resolution on initial mass ejection in
grid-based hydrodynamic simulations of binary neutron star mergers. The subset
of the dynamical ejecta with velocities greater than $\sim 0.6$c can generate
an ultraviolet precursor to the kilonova on $\sim$hr timescales and contribute
to a years-long non-thermal afterglow. Previous work has found differing
amounts of this fast ejecta, by one- to two orders of magnitude, when using
particle-based or grid-based hydrodynamic methods. Here we carry out a
numerical experiment that models the merger as an axisymmetric collision in a
co-rotating frame, accounting for Newtonian self-gravity, inertial forces, and
gravitational wave losses. The lower computational cost allows us to reach
spatial resolutions as high as $4$m, or $\sim 3\times 10^{-4}$ of the stellar
radius. We find that fast ejecta production converges to within $10\%$ for a
cell size of $20$m. This suggests that fast ejecta quantities found in existing
grid-based merger simulations are unlikely to increase to the level needed to
match particle-based results upon further resolution increases. The resulting
neutron-powered precursors are in principle detectable out to distances
$\lesssim 200$Mpc with upcoming facilities. We also find that head-on
collisions at the free-fall speed, relevant for eccentric mergers, yield fast
and slow ejecta quantities of order $10^{-2}M_\odot$, with a kilonova signature
distinct from that of quasi-circular mergers.
