The Godunov method, is adapted to computation of dambreak flood waves. Review of commonly used methods for discontinuous flows (flows with shocks) has shown the need for a new type of algorithm to cope efficiently with mixed flow regimes and strong shocks in non-prismatic channels. The one-dimensional open-channel flow equations are solved with two variants of the Godunov method: one based on linear, the other on piecewise parabolic interpolation (PPM). The proposed methods agree well with the analytical solution (where available), and perform significantly better than the other compared methods for all examined cases.