  The present work considers diffusive shock acceleration at non-relativistic
shocks using a system of stochastic differential equations (SDE) equivalent to
the Fokker-Planck equation. We compute approximate solutions of the transport
of cosmic particles at shock fronts with a SDE numerical scheme. The momentum
gain is given by implicit calculations of the fluid velocity gradients using a
linear interpolation between two consecutive time steps. We validate our
procedure in the case of single shock acceleration with different shock
thickness, with or without synchrotron losses. A comparative discussion of
implicit and explicit schemes for different shock thickness shows that implicit
calculations extend the range of applicability of SDE schemes to infinitely
thin 1D shocks. The method is then applied to multiple shock acceleration for a
system of identical shocks which free parameters are the distance between two
consecutive shocks, the synchrotron losses time and the escape time of the
particles. The stationary distribution reproduces quite well the flat
differential logarithm energy distribution produced by multiple shock effect,
and also the piling-up effect due synchrotron losses at a momentum where they
equilibrate the acceleration rate. At higher momenta particle losses dominate
and the spectrum drops. The competition between acceleration and loss effects
leads to a pile-up shaped distribution which appears to be effective only in a
restrict range of inter-shock distances of 10-100 diffusion lengths. We finally
compute the optically thin synchrotron spectrum produced such periodic pattern
which can explain flat and/or inverted spectra observed in Flat Radio spectrum
Quasars and in the galactic centre.
