## 5. Conclusions and outlookAchterberg & Krülls (1992) and Krülls & Achterberg (1994, KA94) have demonstrated the usefulness of the stochastic differential equation approach as an efficient tool for investigating particle acceleration at shock fronts. They found approximate solutions of the diffusion-advection equations in several different physical situations including second order Fermi acceleration and momentum dependent spatial diffusion coefficients. However, their computational approach is limited to spatially resolved shock structures, and needs unrealistically short time steps for thin shocks. We have presented an improved scheme in which the particle momentum gain during a time step depends on both the initial and final positions of the particle. The scheme reproduces analytical results derived for shock thicknesses much lower than a typical diffusive length, without imposing an excessive requirement on the time step. We have applied our procedure to a system of multiple identical shocks. The signature is obtained when the escape time is much larger than the multiple shock acceleration time. Inclusion of losses leads to a pile-up effect where the acceleration and loss rates are equal, as has been found analytically. The pile-up is present only for a restricted range of parameters: when the escape time exceeds the time for the plasma to move from one shock to the next, but is short enough not to permit synchrotron cooling of the lowest energy particles. We have shown that parameter ranges can be found in which the spectrum simultaneously displays the power law index characteristic of multiple shock acceleration (at low momenta) and the index appropriate to single shock acceleration at high momenta. The synchrotron spectrum produced in each flow pattern can show flat or inverted spectra depending on the inter-shock distance, the loss strength and the ratio of escape to acceleration time. Multiple shock acceleration may explain the radio spectrum from flat radio Quasars, the radio to IR spectrum of galactic sources such as Sagittarus , and even hard X-ray spectra observed in high states of the BL Lac Markarian 501. Several extensions of this work are possible, such as the consideration of non-periodic flows and non-stationary injection. It also seems feasible to extend the 1D simulations to more complex multi-dimensional flows, such as numerical simulations of jets. © European Southern Observatory (ESO) 1999 Online publication: June 18, 1999 |