In this paper, various modern characteristics-based high resolution nonoscillatory shock-capturing finite difference and Petrov-Galerkin finite element methods are presented for the computation of one-dimensional unsteady free surface flows resulting from a dam-breaking. The adopted schemes come directly from gas flow computations in gas dynamics. The computed results are compared with analytic solutions to assess their validity. It is demonstrated that accurate predictions of high speed open channel flows are obtainable using the present schemes. An analytic solution for sudden formation of a bore wave is also included.