  Recently, there has been a significant level of discussion of the correct
treatment of Kelvin-Helmholtz instability in the astrophysical community. This
discussion relies largely on how the KHI test is posed and analyzed. We pose a
stringent test of the initial growth of the instability. The goal is to provide
a rigorous methodology for verifying a code on two dimensional Kelvin-Helmholtz
instability. We ran the problem in the Pencil Code, Athena, Enzo, NDSPHMHD, and
Phurbas. A strict comparison, judgment, or ranking, between codes is beyond the
scope of this work, though this work provides the mathematical framework needed
for such a study. Nonetheless, how the test is posed circumvents the issues
raised by tests starting from a sharp contact discontinuity yet it still shows
the poor performance of Smoothed Particle Hydrodynamics. We then comment on the
connection between this behavior to the underlying lack of zeroth-order
consistency in Smoothed Particle Hydrodynamics interpolation. We comment on the
tendency of some methods, particularly those with very low numerical diffusion,
to produce secondary Kelvin-Helmholtz billows on similar tests. Though the lack
of a fixed, physical diffusive scale in the Euler equations lies at the root of
the issue, we suggest that in some methods an extra diffusion operator should
be used to damp the growth of instabilities arising from grid noise. This
statement applies particularly to moving-mesh tessellation codes, but also to
fixed-grid Godunov schemes.
