  [...] We present results for the statistics of thermal gas and the shock wave
properties for a large volume simulated with three different cosmological
numerical codes: the Eulerian total variations diminishing code TVD, the
Eulerian piecewise parabolic method-based code ENZO, and the Lagrangian
smoothed-particle hydrodynamics code GADGET. Starting from a shared set of
initial conditions, we present convergence tests for a cosmological volume of
side-length 100 Mpc/h, studying in detail the morphological and statistical
properties of the thermal gas as a function of mass and spatial resolution in
all codes. By applying shock finding methods to each code, we measure the
statistics of shock waves and the related cosmic ray acceleration efficiencies,
within the sample of simulations and for the results of the different
approaches. We discuss the regimes of uncertainties and disagreement among
codes, with a particular focus on the results at the scale of galaxy clusters.
Even if the bulk of thermal and shock properties are reasonably in agreement
among the three codes, yet some significant differences exist (especially
between Eulerian methods and smoothed particle hydrodynamics). In particular,
we report: a) differences of huge factors (10-100) in the values of average gas
density, temperature, entropy, Mach number and shock thermal energy flux in the
most rarefied regions of the simulations between grid and SPH methods; b) the
hint of an entropy core inside clusters simulated in grid codes; c)
significantly different phase diagrams of shocked cells in grid codes compared
to SPH; d) sizable differences in the morphologies of accretion shocks between
grid and SPH methods.
