Numerical Cosmic-Ray Hydrodynamics

We present a numerical method for integrating the equations describing a system made of a fluid and cosmic-rays. We work out the modified characteristic equations that include the CR dynamical effects in smooth flows. We model the energy exchange between cosmic-rays and the fluid, due to diffusive processes in configuration and momentum space, with a flux conserving method. For a specified shock acceleration efficiency as a function of the upstream conditions and shock Mach number, we modify the Riemann solver to take into account the cosmic-ray mediation at shocks without resolving the cosmic-ray induced substructure. A self-consistent time-dependent shock solution is obtained by using our modified solver with Glimm's method. Godunov's method is applied in smooth parts of the flow.