An appropriate numerical modeling of cosmic-ray transport is a major challenge in modern astroparticle physics. Multi-messenger observations of cosmic rays and their neutral secondary particles like neutrinos and gamma-rays must relate to source models. This requires a detailed description of the propagation and interaction of all particles.
Especially in the region between the cosmic-ray knee and ankle, where the transition between Galactic and extra-galactic sources is expected, the modeling of cosmic-ray transport is complicated. Classical simulation frameworks are usually dedicated to lower or to the very highest energies. These software tools use grid-based methods to solve the transport equation (low energies) or propagate single particles solving the equation of motion (highest energies). Both techniques having different problems in the transition region.
In this work an innovative technique, based on stochastic differential equations, to solve the transport equation is explained. The ansatz of propagating independent phase-space elements (or pseudo-particles) allows for easy parallelization which makes it attractive for modern large-scale computing architectures. It combines the numerical framework of the single particle propagation with the advantage of ensemble averaged description of the transport equation. Making it an ideal tool to describe the transition between Galactic and extra-galactic cosmic-rays.
This approach is implemented in the latest version of the publicly available propagation software CRPropa and already applied to various problems. The influence of different source distributions on the global Galactic cosmic-ray density and the transport of cosmic-rays from the Galactic Wind Termination shock are discussed as an example.