  We model the diffusive shock acceleration of particles in a system of two
colliding shock waves and present a method to solve the time-dependent problem
analytically in the test-particle approximation and high energy limit. In
particular, we show that in this limit the problem can be analysed with the
help of a self-similar solution. While a number of recent works predict hard
($E^{-1}$) spectra for the accelerated particles in the stationary limit, or
the appearance of spectral breaks, we found instead that the spectrum of
accelerated particles in a time-dependent collision follows quite closely the
canonical $E^{-2}$ prediction of diffusive shock acceleration at a single
shock, except at the highest energy, where a hardening appears, originating a
bumpy feature just before the exponential cutoff. We also investigated the
effect of the reacceleration of pre-existing cosmic rays by a system of two
shocks, and found that under certain conditions spectral features can appear in
the cutoff region. Finally, the mathematical methods presented here are very
general and could be easily applied to a variety of astrophysical situations,
including for instance standing shocks in accretion flows, diverging shocks,
backward collisions of a slow shock by a faster shock, and wind-wind or
shock-wind collisions.

High Energy Astrophysical Phenomena