## 1. Introduction
The non-thermal radio and high-energy radiation spectra from
extragalactic and galactic objects such as X-ray binaries,
micro-quasars, active galactic nuclei, jets and gamma-ray bursts
require the presence of non-thermal particle distributions. One of the
prime mechanisms for producing such distributions is diffusive
acceleration at a shock front. This theory assumes the particle
distributions are isotropised by efficient scattering on wave
turbulence on both sides of the shock and gain energy by a first order
Fermi process upon crossing it. The equation governing the particle
distribution is of the Fokker-Planck type, containing both advection
(dynamical friction) and diffusion terms in the spatial variables and
an advection term in the energy variable (or in the magnitude of the
particle momentum It is only recently that SDE systems have been applied to astrophysical problems: in solar physics with the investigation of acceleration of fast electrons in the solar corona (MacKinnon & Craig 1991, Conway et al. 1998) and in space physics with ion acceleration at the solar termination shock (Chalov et al. 1995). In two papers Achterberg & Krülls (1992) and Krülls & Achterberg (1994 - henceforth KA94), have applied the SDE approach to the problem of particle acceleration at astrophysical shocks, including the possibility of second-order Fermi acceleration (Schlickeiser 1989). These authors drew several important conclusions. In particular, they showed that for Kolmogorov turbulence the second-order acceleration effect is restricted to a small momentum range close to that of injection, because of the increase of the diffusion time with momentum. At high momentum, in contrast, the spectrum is formed by the competing effects of synchrotron losses and the first-order Fermi process. The numerical scheme used in KA94 is The organisation of the paper is as follows: in Sect. 2 we present the SDE system equivalent to the advection-diffusion equation and describe the explicit and implicit integration methods. We then test the implicit scheme in Sect. 3 by applying it to the (well-known) problem of acceleration at a single isolated shock front, both with and without synchrotron losses. We also compare the performances of implicit and explicit schemes. Sect. 4 focuses on the application to acceleration in a system of multiple shocks. Here, as well as testing the code against known approximate solutions, we present new solutions valid in regions of parameter space inaccessible to the analytic methods. We use the results to calculate the optically thin synchrotron radiation produced by the stationary particle distribution within a periodic pattern of shocks. In Sect. 5 we summarise our results and discuss extensions of the method to more complex problems. © European Southern Observatory (ESO) 1999 Online publication: June 18, 1999 |