  Shock acceleration is considered one of the most important mechanisms for the
acceleration of astrophysical energetic particles. In this work, we calculate
the trajectories of a large number of test charged particles accurately in a
parallel shock with magnetic turbulence. We investigate the time evolution of
the accelerated-particle energy spectrum in the downstream of the shock in
order to understand the acceleration mechanism of energetic particles. From
simulation results we obtain power-law energy spectra with a bend-over energy,
$E_0$, increasing with time. With the particle mean acceleration time and mean
momentum change during each cycle of the shock crossing from diffusive shock
acceleration model (following Drury), a time-dependent differential equation
for the maximum energy, $E_{acc}$, of particles accelerated at the shock, can
be approximately obtained. We assume the theoretical bend-over energy as
$E_{acc}$. It is found that the bend-over energy from simulations agrees well
with the theoretical bend-over energy using the non-linear diffusion theory,
NLGCE-F, in contrast to that using the classic quasi-linear theory (QLT).
