ner_tags: O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O B-Identifier I-Identifier B-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software O O O O B-Software O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O B-Software I-Software I-Software O B-Software I-Software O O O O O O O B-Software I-Software I-Software I-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software
tokens: in hadronic scenario, with spectrum and emissivity directly related to the spectrum of highest energy CRs. The simplified model of gamma-ray evolution, used in our paper, is not strongly influenced by the shape of the spectrum as it approximates it with E 2 . 2.1 Hydrodynamic modelling The dynamical evolution of an SNR was modelled by numerically solving the time-dependent HD equations of mass, momentum and energy conservation: where is the mass density, is the flow velocity, P is the thermal pressure, is the unit vector, E is the total energy density and is the adiabatic index. The total energy density is the sum of the thermal and kinetic components: We adopt the pluto code (Version 4.2; Mignone etal. 2007 , 2012 ) to solve the system of HD conservation laws by using a cell-centred finite-volume approach based on Godunov-type schemes. The code design enables efficient usage of massively parallel computers through the message passing interface standard (MPI) for interprocessor communications. We do not include radiative cooling and thermal conduction in our HD equations, describing only the free and Sedov expansion phases of the SNR evolution in a tenuous, collisionless medium. The transition time from Sedov to radiative phase for an SNR is described with the approximation (e.g. Blondin etal. 1998 ; Petruk 2005 ; Orlando etal. 2011 ) where E 51 E 0 /(10 51 erg) and E 0 is the initial total explosion energy, contained mostly in the form of kinetic energy. In our set of simulations, t tr 0.5Myr and therefore our modelled SNRs never reach the radiative phase. We will compute the radio emission from the SNR but with assumption that the radiation has no impact yet on its dynamical evolution. Throughout our modelling, we do not activate the MHD solver of pluto because it is in any case powerless in describing the generation of magnetic turbulence by CRs upstream of the shock and corresponding MFA. Such an amplified magnetic field is dominant compared to the field compressed only due to fluid compression, especially for young SNRs where non-linearity is very pronounced. NLDSA module, which runs parallel with pluto HD code, simultaneously performs calculations of MFA, synthesize the global radio emission in this amplified field and also accounts for its impact on hydrodynamics. We used the following set of pluto algorithms in our simulations: linear interpolation with default limiter, HLLC Riemann solver and RK2 algorithm for the time evolution. Additionally, we used MULTID shock flattening algorithm for the numerical dissipation near the strong shocks. Our three-dimensional (3D) computations were carried out in spherical coordinates ( r , , ), by using a static logarithmic grid, with mesh size increasing with the SNR radius. Detection and tracking of the SNR shock waves in the fluid, travelling in some direction x , is based on two standard numerical conditions, namely and , where p represents parameter, setting the shock strength. We modified pluto