Two new finite-difference schemes - Gabutti, and Beam and Warming - are introduced and compared for the analysis of unsteady free-surface flows resulting from the breaking of a dam. These schemes split the fluxvector into positive and negative parts, each of which corresponds to the direction of a characteristic, thereby allowing use of proper finite differences for the space derivatives. Central finite differences are used for subcritical flow and upwind differences are used for supercritical flow. The details of these schemes are presented and the computed results are compared with the analytical solution to demonstrate their validity. Because of their simplicity, these schemes are attractive for solving the dam-break problem, especially when supercritical flow is present.