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Cosmic rays (CRs) are frequently modeled as an additional fluid inhydrodynamic (HD) and magnetohydrodynamic (MHD) simulations of astrophysicalflows. The standard CR two-fluid model is described in terms of threeconservation laws (expressing conservation of mass, momentum and total energy)and one additional equation (for the CR pressure) that cannot be cast in asatisfactory conservative form. The presence of non-conservative terms withspatial derivatives in the model equations prevents a unique weak solutionbehind a shock. We investigate a number of methods for the numerical solutionof the two-fluid equations and find that, in the presence of shock waves, theresults generally depend on the numerical details (spatial reconstruction, timestepping, the CFL number, and the adopted discretization). All methods convergeto a unique result if the energy partition between the thermal and non-thermalfluids at the shock is prescribed using a subgrid prescription. This highlightsthe non-uniqueness problem of the two-fluid equations at shocks. From ournumerical investigations, we report a robust method for which the solutions areinsensitive to the numerical details even in absence of a subgrid prescription,although we recommend a subgrid closure at shocks using results from kinetictheory. The subgrid closure is crucial for a reliable post-shock solution andalso its impact on large scale flows because the shock microphysics thatdetermines CR acceleration is not accurately captured in a fluid approximation.Critical test problems, limitations of fluid modeling, and future directionsare discussed.

A numerical approach to the non-uniqueness problem of cosmic ray two-fluid equations at shocks