Incorporation of cosmic ray transport into the ZEUS MHD code. Application for studies of Parker instability in the ISM

We present a numerical algorithm for the incorporation of the active cosmic ray transport into the ZEUS-3D magnetohydrodynamical code. The cosmic ray transport is described by the diffusion-advection equation. The applied form of the diffusion tensor allows for anisotropic diffusion of cosmic rays along and across the magnetic field direction, which is controlled by two parameters: the parallel and perpendicular diffusion coefficients. The implemented numerical algorithm is tested by comparison of the diffusive transport of cosmic rays to analytical solutions of the diffusion equation. Our method is numerically stable for a wide range of diffusion coefficients, including the realistic values inferred from the observational data for the Milky Way of about 6 x 10²⁸ cm² s^-1. The presented algorithm is applied for exemplary simulations of the Parker instability triggered by cosmic rays injected by a single SN remnant.