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Context. The diffusive shock acceleration mechanism has been widely acceptedas the acceleration mechanism for galactic cosmic rays. While self-consistenthybrid simulations have shown how power-law spectra are produced, detailedinformation on the interplay of diffusive particle motion and the turbulentelectromagnetic fields responsible for repeated shock crossings are stillelusive. Aims. The framework of test-particle theory is applied to investigatethe effect of diffusive shock acceleration by inspecting the obtainedcosmic-ray energy spectra. The resulting energy spectra can be obtained thisway from the particle motion and, depending on the prescribed turbulence model,the influence of stochastic acceleration through plasma waves can be studied.Methods. A numerical Monte-Carlo simulation code is extended to includecollisionless shock waves. This allows one to trace the trajectories of testparticle while they are being accelerated. In addition, the diffusioncoefficients can be obtained directly from the particle motion, which allowsfor a detailed understanding of the acceleration process. Results. The classicresult of an energy spectrum with $E^{-2}$ is only reproduced for parallelshocks, while, for all other cases, the energy spectral index is reduceddepending on the shock obliqueness. Qualitatively, this can be explained interms of the diffusion coefficients in the directions that are parallel andperpendicular to the shock front.

Cosmic-ray acceleration at collisionless astrophysical shocks using Monte-Carlo simulations