  In the interstellar medium of galaxies and the intracluster gas of galaxy
clusters, the charged particles making up cosmic rays are moving almost
exclusively along (but not across) magnetic field lines. The resulting
anisotropic transport of cosmic rays in the form of diffusion or streaming not
only affects the gas dynamics but also rearranges the magnetic fields
themselves. The coupled dynamics of magnetic fields and cosmic rays can thus
impact the formation and evolution of galaxies and the thermal evolution of
galaxy clusters in critical ways. Numerically studying these effects requires
solvers for anisotropic diffusion that are accurate, efficient, and robust,
requirements that have proven difficult to satisfy in practice. Here, we
present an anisotropic diffusion solver on an unstructured moving mesh that is
conservative, does not violate the entropy condition, allows for semi-implicit
time integration with individual timesteps, and only requires solving a single
linear system of equations per timestep. We apply our new scheme to a large
number of test problems and show that it works as well or better than previous
implementations. Finally, we demonstrate for a numerically demanding simulation
of the formation of an isolated disk galaxy that our local time-stepping scheme
reproduces the results obtained with global time-stepping at a fraction of the
computational cost.
