  We compare the statistics of driven, supersonic turbulence at high Mach
number using FLASH a widely used Eulerian grid-based code and PHANTOM, a
Lagrangian SPH code at resolutions of up to 512^3 in both grid cells and SPH
particles. We find excellent agreement between codes on the basic statistical
properties: a slope of k^-1.95 in the velocity power spectrum for hydrodynamic,
Mach 10 turbulence, evidence in both codes for a Kolmogorov-like slope of
k^-5/3 in the variable rho^1/3 v as suggested by Kritsuk et al. and a
log-normal PDF with a width that scales with Mach number and proportionality
constant b=0.33-0.5 in the density variance-Mach number relation. The measured
structure function slopes are not converged in either code at 512^3 elements.
We find that, for measuring volumetric statistics such as the power spectrum
slope and structure function scaling, SPH and grid codes give roughly
comparable results when the number of SPH particles is approximately equal to
the number of grid cells. In particular, to accurately measure the power
spectrum slope in the inertial range, in the absence of subgrid models,
requires at least 512^3 computational elements in either code. On the other
hand the SPH code was found to be better at resolving dense structures, giving
max. densities at a resolution of 128^3 particles that were similar those
resolved in the grid code at 512^3 cells, reflected also in the high density
tail of the PDF. We find SPH to be more dissipative at comparable numbers of
computational elements in statistics of the velocity field, but correspondingly
less dissipative than the grid code in the statistics of density weighted
quantities such as rho^1/3 v. For SPH simulations of high Mach number
turbulence we find it important to use sufficient non-linear beta-viscosity to
prevent particle interpenetration in shocks (we require beta=4 instead of the
default beta=2).
