[en] The Piecewise-Parabolic Method (PPM) has been used to study the nonlinear development of the Kelvin-Helmholtz instability of a Mach 2 slip surface in both a gamma-law gas and in an isothermal gas. A simplified version of PPM appropriate to this and other problems with only weak shocks is described. The instability calculations demonstrate the usefulness of discontinuity steepening in PPM and they illustrate the complexity in a flow problem which this method can treat accurately on Cray-I-class computers. The simulations also bring to light characteristic combinations of nonlinear waves which arise from finite-amplitude perturbations of the slip surface and which exhibit an approximately self-similar growth. After passing through a fairly chaotic phase of development, the mixing layer generated by the instability achieves a relatively ordered state which does not appear to depend greatly upon the nature of the initial perturbation, but which does depend upon the length scale over which strict periodicity is enforced in the simulation