Ellison et al. (2005) compare a recent semi-analytic model of non-linear diffusive shock acceleration (Blasi, 2002, 2004; Blasi et al., 2005) with a well-established Monte Carlo model (e.g., Ellison et al., 1990, 1999; Jones and Ellison, 1991). Both include a thermal leakage model for injection and the non-linear back reaction of accelerated particles on the shock structure, but they do these in very different ways. Also different is the way in which the particle diffusion in the background magnetic turbulence is modeled. The limited comparison of Ellison et al. (2005) shows that the important non-linear effects of compression ratios >> 4 and concave spectra do not depend strongly on the injection model as long as injection is efficient. A fuller understanding of the complex plasma processes involved, particularly if injection is weak, will require particle-in-cell simulations (e.g., Giacalone and Ellison, 2000), but these simulations, which must be done fully in three-dimensions (Jones et al., 1998), cannot yet be run long enough, in large enough simulation spaces, to accelerate particles from thermal to relativistic energies in order to show strong non-linear effects. For now, approximate methods must be used. The Monte Carlo model is more general than the semi-analytic model. For instance, Ellison et al. (2005) note, that the Monte Carlo model can treat a specific momentum dependence for the scattering mean free path, particle acceleration in relativistic shocks (Ellison and Double, 2002, 2004), and non-linear effects in oblique shocks (Ellison et al., 1999), but it is considerably slower computationally. Since it is important in many applications, such as hydro models of supernova remnants, to include the dynamic effects of nonlinear diffusive shock acceleration in simulations which perform the calculation many times (e.g., Ellison et al., 2004), a rapid, approximate calculation, such as the semi-analytic model discussed here, is useful.
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