The unsteady, one-dimensional Saint-Venant equations are solved by an implicit finite difference scheme to handle general channel and river flows. The initial conditions for the unsteady flow are provided by solving the steady varied flow equation for the specified boundary conditions. The solution for the unsteady flow allows any of eight separate boundary conditions to be specified which are composed of combinations of specifying the depth or discharge as functions of time at either the upstream of downstream ends, with the stage-discharge relation or constant depth and flow rate specified at the other end. Typical solutions showing the spatial and time dependency of such flow characteristics as flow rate, depth and velocity are given for example problems, which include lateral inflow, and channels whose geometry, slope and Manning’s n vary with respective to distance along the channel.
Jeppson, Roland W., "Simulation of Steady and Unsteady Flows in Channels and Rivers" (1974). Reports. Paper 301.