  (abridged) We apply a second-order Godunov code, Athena, to studies of the
magnetorotational instability using unstratified shearing box simulations with
a uniform net vertical field and a sinusoidally varying zero net vertical
field. The Athena results agree well with similar studies that used different
numerical algorithms. We conduct analyses to study the flow of energy from
differential rotation to turbulent fluctuations to thermalization. A study of
the temporal correlation between the time derivatives of volume-averaged energy
components shows that energy injected into turbulent fluctuations dissipates on
a timescale of $\Omega^{-1}$, where $\Omega$ is the orbital frequency of the
local domain. Magnetic dissipation dominates over kinetic dissipation, although
not by as great a factor as the ratio of magnetic to kinetic energy. We
Fourier-transform the magnetic and kinetic energy evolution equations and,
using the assumption that the time-averaged energies are constant, determine
the level of numerical dissipation as a function of length scale and
resolution. By modeling numerical dissipation as if it were physical in origin,
we characterize numerical resistivity and viscosity in terms of effective
Reynolds and Prandtl numbers. The resulting effective magnetic Prandtl number
is $\sim 2$, independent of resolution or initial field geometry. MRI
simulations with effective Reynolds and Prandtl numbers determined by numerical
dissipation are not equivalent to those where these numbers are set by physical
resistivity and viscosity. These results serve, then, as a baseline for future
shearing box studies where dissipation is controlled by the inclusion of
explicit viscosity and resistivity.
