The origin of high-energy cosmic rays (CRs) in the transition region between the knee and the ankle is still debated. In general, CRs are most likely accelerated stochastically in time-dependent, turbulent magnetic field structures. Diffusive Shock Acceleration at stationary shocks produce the characteristic power-law spectrum with the spectral slope depending only on the shock's compression ratio. The spectrum that is observed downstream can be modulated by properties of diffusive transport.
How do a finite shock width and diffusion properties influence the time evolution of the spectrum? And, how does it change when cosmic rays are already pre-accelerated to a power-law spectrum when they enter the acceleration region?

We assess those questions using the stochastic differential equation solver (DiffusionSDE) of the cosmic-ray propagation framework CRPropa3.2. Assuming continuous injection of cosmic rays in the acceleration region, the time evolution of the spectrum at a spherical shock is obtained for energy-independent and energy-dependent diffusion. We show that the energy-spectrum at the shock may be steeper than the ideal shock spectrum. The injection of pre-accelerated cosmic rays can lead to a broken power-law spectrum.

We apply our findings to the re-acceleration of cosmic rays at the Galactic Wind Termination Shock (GWTS). First results of modelling a spherically symmetric GWTS and a spiral Galactic magnetic field are presented. We conclude that time-resolved simulations are necessary to constrain the contribution of GWTS to the CR flux, considering a finite shock lifetime, finite shock width, energy-dependent diffusion, complex magnetic field and upstream cooling.