Glimm–Godunov’s method for cosmic-ray-hydrodynamics

Abstract

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.

Modelling injection and feedback of cosmic rays in grid-based cosmological simulations: effects on cluster outskirts

“…In this work, we consider only CR protons , for which to first approximation radiative and Coulomb losses and the inelastic proton–proton collisions can be neglected, for the time‐scales of interest here (e.g. Miniati 2007). The extension to CR electrons and the modelling of radiative, Coulomb losses and proton–proton collisions for both species will be investigated in future work.…”

“…Furthermore, when the ‘left’ and ‘right’ edge states of the Riemann problem are computed for the solution of the fluxes, the appropriate wave velocity is used; this implies replacing with in the associated routines of the PPM algorithm, as suggested in Miniati (2007).…”

The impact of the SZ effect on cm-wavelength (1–30 GHz) observations of galaxy cluster radio relics

“…For the former, we assume a standard spherical shock geometry. This is clearly a simplification as simulations show that although shock fronts radiate outward in all direction after a major merger, its geometry is not actually spherical due to the inhomogeneous structure of the ICM (e.g., Miniati 2000). However, for our purposes it suffices that the SZ signal boost is localized within the same geometrical boundary as marked by the relics, because the flux contamination is determined primarily from the projection of pressure from a narrow annular section of the shock where the relic resides.…”

“…The second point is more complicated, namely the width of the shock-boosted region. Lacking any sophisticated simulations for gas cooling, we follow the examples from shock-tube geometry (e.g., Miniati 2007;Vazza et al 2012), simply assuming that the boosted pressure stays constants for a small distance (which we can call the shock width) after which it drops owing to cooling with a power law and merges into the ambient pressure. This setting is shown in the top panel of Fig.…”

Block structured adaptive mesh and time refinement for hybrid, hyperbolic+N-body systems

“…The code is based on the CHOMBO AMR library and it is implemented in a hybrid C++/ Fortran77 language. Additional physics modules, such as radiation [40], cosmic-rays [41,42] and magnetohydrodynamics will be presented elsewhere. In the following, we focus on numerical tests to assess the performance of the algorithms in terms of accuracy and applicability to problems of direct interest.…”