  We examine the accuracy of spatial derivatives computed from numerical
simulations of supersonic turbulence. Two sets of simulations, carried out
using a finite-volume code that evolves the hydrodynamic equations with an
approximate Riemann solver and a finite-difference code that solves the
Navier-Stokes equations, are tested against a number of criteria based on the
continuity equation, including exact results at statistically steady state. We
find that the spatial derivatives in the Navier-Stokes runs are accurate and
satisfy all the criteria. In particular, they satisfy our exact results that
the conditional mean velocity divergence, $\langle \nabla \cdot {\bs
u}|s\rangle$, where $s$ is the logarithm of density, and the conditional mean
of the advection of $s$, $\langle {\bs u} \cdot \nabla s|s\rangle$, vanish at
steady state for all density values, $s$. On the other hand, the Riemann solver
simulations fail all the tests that require accurate evaluation of spatial
derivatives, resulting in apparent violation of the continuity equation, even
if the solver enforces mass conservation. In particular, analysis of the
Riemann simulations may lead to the incorrect conclusion that the $\pdv$ work
tends to preferentially convert kinetic energy into thermal energy,
inconsistent with the exact result that the energy exchange by $\pdv$ work is
symmetric in barotropic supersonic turbulence at steady state. The inaccuracy
of spatial derivatives is a general problem in the post-processing of
simulations of supersonic turbulence with Riemann solvers. Solutions from such
simulations must be used with caution in post-processing studies concerning the
spatial gradients.
