  Eulerian hydrodynamical simulations are a powerful and popular tool for
modeling fluids in astrophysical systems. In this work, we critically examine
recent claims that these methods violate Galilean invariance of the Euler
equations. We demonstrate that Eulerian hydrodynamics methods do converge to a
Galilean-invariant solution, provided a well-defined convergent solution
exists. Specifically, we show that numerical diffusion, resulting from
diffusion-like terms in the discretized hydrodynamical equations solved by
Eulerian methods, accounts for the effects previously identified as evidence
for the Galilean non-invariance of these methods. These velocity-dependent
diffusive terms lead to different results for different bulk velocities when
the spatial resolution of the simulation is kept fixed, but their effect
becomes negligible as the resolution of the simulation is increased to obtain a
converged solution. In particular, we find that Kelvin-Helmholtz instabilities
develop properly in realistic Eulerian calculations regardless of the bulk
velocity provided the problem is simulated with sufficient resolution (a factor
of 2-4 increase compared to the case without bulk flows for realistic
velocities). Our results reiterate that high-resolution Eulerian methods can
perform well and obtain a convergent solution, even in the presence of highly
supersonic bulk flows.
