Time evolution of the downstream ion momentum spectrum for M = 20 parallel shock (Run B; see Table 1), showing both the thermal component (p ≲ 2 mv_sh), and accelerated particles. The non-thermal power-law tail ∝p⁻⁴ agrees with DSA prediction at strong shocks (see Paper I). The maximum momentum increases until , when the diffusion length of the most energetic ions becomes comparable with the box size.