We present in this paper numerical simulations of reactive flows inter
acting with shock waves. We argue that spectral methods are suitable f
or these problems and review the recent developments in spectral metho
ds that have made them a powerful numerical tool appropriate for long-
term integrations of complicated flows, even in the presence of shock
waves. A spectral code is described in detail, and the theory that lea
ds to each of its components is explained. Results of interactions of
hydrogen jets with shock waves are presented and analyzed, and compari
sons with ENO finite difference schemes are carried out.