Glimm-Godunov's method for cosmic-ray-hydrodynamics

A numerical method for integrating the equations describing a dynamically coupled system made of a fluid and cosmic-rays is developed. In smooth flows the effect of CR pressure is accounted for by modification of the characteristic equations and the energy exchange between cosmic-rays and the fluid, due to diffusive processes in configuration and momentum space, is modeled with a flux conserving method. Provided the shock acceleration efficiency as a function of the upstream conditions and shock Mach number, we show that the Riemann solver can be modified to take into account the cosmic-ray mediation without having to resolve the cosmic-ray induced substructure. Shocks are advanced with Glimm’s method which preserves their discontinuous character without any smearing, thus allowing to maintain selfconsistency in the shock solutions. In smooth flows either Glimm’s or a higher order Godunov’s method can be applied, with the latter producing better results when approximations are introduced in the Riemann solver.