2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/1974One of the key questions to understanding the efficiency of diffusive shock acceleration of the cosmic rays (CRs) is the injection process from thermal particles. A self-consistent injection model based on the interactions of the suprathermal particles with self-generated magneto-hydrodynamic waves has been developed recently by Malkov (1998, Phys. Rev. E 58, 4911). By adopting this analytic solution, a numerical treatment of the plasma-physical injection model at a strong quasi-parallel shock has been devised and incorporated into the combined gas dynamics and the CR diffusion-convection code. In order to investigate self-consistently the injection and acceleration efficiencies, we have applied this code to the CR modified shocks of both high and low Mach numbers (M=30 and M=2.24) with a Bohm type diffusion model. We find the injection process is self-regulated in such a way that the injection rate reaches and stays at a nearly stable value after quick initial adjustment. For both shocks about 0.1% of the incoming thermal particles are injected into the CRs. For the weak shock, the shock has reached a steady state within our integration time and about 10% of the total available shock energy is transfered into the CR energy density. The strong shock has achieved a higher acceleration efficiency of about 20% by the end of our simulation, but has not yet reached a steady-state.13 pages, 10 figures, LaTeX209 (epsf.sty, l-aa.sty), Astron.Astrophys. in pressAstrophysicstext