Modeling Cosmic-Ray Acceleration at Nonplanar Astrophysical Shocks Using the SPECTRUM Code

Abstract Diffusive shock acceleration is an established mechanism for accelerating charged particles to high energies, both inside the heliosphere and beyond its boundaries. In this paper, we rigorously test and benchmark the Space Plasma and Energetic Charged particle TRansport on Unstructured Meshes (SPECTRUM) open-source software on steady-state and time-dependent problems for planar and spherical shock waves based on Parker’s cosmic-ray transport equation. In particular, we investigate the accuracy of the SPECTRUM algorithm as a function of the shock width, the time step, and other simulation parameters. We compare the forward-in-time and backward-in-time methods for simulating stochastic trajectories, proving that both approaches are valid and yield correct results by benchmarking them against analytic solutions. We also compare source-term and boundary-condition methods to normalize the distribution function for problems with and without distributed energy losses. In each verified case, the simulation results match the analytic prediction, demonstrating the suitability of SPECTRUM to study cosmic-ray shock acceleration in diverse space plasma environments.