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Abstract

The time-dependent process of diffusive shock acceleration at astrophysical shocks is investigated in this thesis. Using the Vlasov-Fokker-Planck equation as a foundation, analytical predictions for the time dependence of acceleration and particle transport are derived. The impact of shock thickness on the spectral index is examined, an important step in explaining discrepancies between observational data and theoretical predictions. The particle transport code Sapphire++ was used, which solves the Vlasov-Fokker-Planck equation with high accuracy using a discontinuous Galerkin method. Its accuracy was tested against analytical solutions, showing deviations below 0.5%. My results for both momentum-independent and momentum-dependent scattering mark a step towards a deeper understanding of the dynamics of diffusive shock acceleration. The simulations revealed that finite shock thickness influences the spectral index, with wider shocks leading to steeper spectra. The thesis is focused on parallel shocks, but lays the groundwork for exploring more complex configurations, such as oblique shocks and spatially varying scattering. The results highlight Sapphire++ as a valuable resource for future research into cosmic ray acceleration and the physics of astrophysical shocks.