Summary: We present the piecewise parabolic method, a higher-order extension of Godunov’s method. There are several new features of this method which distinguish it from other higher-order Godunov-type methods. We use a higher-order spatial interpolation than previously used, which allows for a steeper representation of discontinuities, particularly contact discontinuities. We introduce a simpler and more robust algorithm for calculating the nonlinear wave interactions used to compute fluxes. Finally, we recognize the need for additional dissipation in any higher- order Godunov method of this type, and introduce it in such a way so as not to degrade the quality of the results.