This is sufficient to account for the CR energy replenishment rate of erg , required by diffusive escape from the galaxy. In DSA theory charged particles gain energy by repeated pitch angle scatterings across the velocity jump of the shock. The scattering MHD waves can be excited by the high energy particles themselves.
Several complications arise.
The total density jump through the shock can exceed the nominal factor of 4 expected in a strong, adiabatic gas shock. Then CRs with different scattering lengths will, on average, encounter different velocity jumps. Earlier, numerical simulations based on nonlinear Monte Carlo techniques, for example, by Ellison et al. The amount of energy held by CRs at different evolutionary stages of a SNR depends on the shock speed, adiabatic losses and the rate of increase of the swept-up matter. The full, nonlinear DSA problem is difficult to compute, especially as applied to SNRs, where shocks are neither steady nor planar.