Diffusive shock acceleration (Fermi acceleration) of cosmic rays in a system containing two shock waves is investigated using test particle simulations and analysis of the diffusion convection equation. We assume that the cosmic ray acceleration timescale is much less than the approaching timescale of the two shocks. Low-energy cosmic rays are primarily accelerated as they interact with one of the two shocks. However, as they are energized, and their mean free path becomes comparable to the separation distance of the two shocks, they start to accelerate as they interact with both shocks. As a result, a double power-law-type spectrum for the cosmic ray distribution is produced. The numerical result is well explained by a model based on the diffusion convection equation, which accounts for the dependence of the diffusion coefficient on the cosmic ray energy. The break point of the spectrum can be estimated by considering transport of cosmic rays from one shock to the other.[Figure not available: see fulltext.].
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