Spectrally resolved cosmic ray hydrodynamics -- I. Spectral scheme
  Cosmic ray (CR) protons are an important component in many astrophysical
systems. Processes like CR injection, cooling, adiabatic changes as well as
active CR transport through the medium strongly modify the CR momentum
distribution and have to be taken into account in hydrodynamical simulations.
We present an efficient novel numerical scheme to accurately compute the
evolution of the particle distribution function by solving the Fokker-Planck
equation with a low number of spectral bins (10 - 20), which is required to
include a full spectrum for every computational fluid element. The distribution
function is represented by piecewise power laws and is not forced to be
continuous, which enables an optimal representation of the spectrum. The
Fokker-Planck equation is solved with a two-moment approach evolving the CR
number and energy density. The low numerical diffusion of the scheme reduces
the numerical errors by orders of magnitude in comparison to classical schemes
with piecewise constant spectral representations. With this method not only the
spectral evolution of CRs can be computed accurately in magnetohydrodynamic
simulations but also their dynamical impact as well as CR ionisation. This
allows for more accurate models for astrophysical plasmas, like the
interstellar medium, and direct comparisons with observations.
