  Motivated by cosmic ray (CR) re-acceleration at a potential Galactic Wind
Termination Shock (GWTS), we present a numerical model for time-dependent
Diffusive Shock Acceleration (DSA). We use the stochastic differential equation
solver (DiffusionSDE) of the cosmic ray propagation framework CRPropa3.2 with
two modifications: An importance sampling module is introduced to improve
statistics at high energies in order to keep the simulation time short. An
adaptive time step is implemented in the DiffusionSDE module. This ensures to
efficiently meet constraints on the time and diffusion step, which is crucial
to obtain the correct shock spectra. The time evolution of the spectrum at a
one-dimensional planar shock is verified against the solution obtained by the
grid-based solver VLUGR3 for both energy-independent and energy-dependent
diffusion. We show that the injection of pre-accelerated particles can lead to
a broken power law spectrum in momentum if the incoming spectrum of CRs is
harder than the re-accelerated spectrum. If the injected spectrum is steeper,
the shock spectrum dominates at all energies. We finally apply the developed
model to the GWTS by considering a spherically symmetric shock, a spiral
Galactic magnetic field, and anisotropic diffusion. The time-dependent spectrum
at the shock is modeled as a basis for further studies.
