  Non-thermal (relativistic) particles are injected into the cosmos by
structure formation shock waves, active galactic nuclei and stellar explosions.
We present a suite of unigrid cosmological simulations (up to $2048^3$) using a
two-fluid model in the grid code ENZO. The simulations include the dynamical
effects of cosmic-ray (CR) protons and cover a range of theoretically motivated
acceleration efficiencies. For the bulk of the cosmic volume the modelling of
CR processes is rather stable with respect to resolution, provided that a
minimum (cell) resolution of $\approx 100 ~\rm kpc/h$ is employed. However, the
results for the innermost cluster regions depend on the assumptions for the
baryonic physics. Inside clusters, non-radiative runs at high resolution tend
to produce an energy density of CRs that are below available upper limits from
the FERMI satellite, while the radiative runs are found to produce a higher
budget of CRs. We show that weak ($M \leq 3-5$) shocks and shock-reacceleration
are crucial to set the level of CRs in the innermost region of clusters, while
in the outer regions the level of CR energy is mainly set via direct injection
by stronger shocks, and is less sensitive to cooling and feedback from active
galactic nuclei and supernovae.
