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We present a new numerical algorithm to solve the recently derived equationsof two-moment cosmic ray hydrodynamics (CRHD). The algorithm is implemented asa module in the moving mesh Arepo code. Therein, the anisotropic transport ofcosmic rays (CRs) along magnetic field lines is discretised using apath-conservative finite volume method on the unstructured time-dependentVoronoi mesh of Arepo. The interaction of CRs and gyroresonant Alfv\'en wavesis described by short-timescale source terms in the CRHD equations. We employ acustom-made semi-implicit adaptive time stepping source term integrator toaccurately integrate this interaction on the small light-crossing time of theanisotropic transport step. Both the transport and the source term integrationstep are separated from the evolution of the magneto-hydrodynamical equationsusing an operator split approach. The new algorithm is tested with a variety oftest problems, including shock tubes, a perpendicular magnetised discontinuity,the hydrodynamic response to a CR overpressure, CR acceleration of a warmcloud, and a CR blast wave, which demonstrate that the coupling between CR andmagneto-hydrodynamics is robust and accurate. We demonstrate the numericalconvergence of the presented scheme using new linear and non-linear analyticsolutions.

A finite volume method for two-moment cosmic-ray hydrodynamics on a moving mesh