A numerical solution technique was developed to solve the one-dimensional, partial differential equations of unsteady flow for an upstream solution. The x-t, distance-time, plane was replaced by a rectangular grid of points at which the values of the variables, discharge and depth, were computed using an implicit, finite difference scheme. A discharge hydrograph and rating curve supplied the initial values of discharge and depth along the downstream column of grid points. Boundary values of discharge were known along the top row and bottom row of grid points from steady flow conditions.

The solution proceeded from column to column in the negative x direction yielding an upstream discharge hydrograph as the final solution.

The computer model was successfully tested utilizing a reach of the James River, Virginia. The upstream hydrograph was first routed in the downstream direction using an implicit solution procedure already available.

The downstream hydrograph resulting as a solution was used as input to the reverse flood routing model. The upstream solution as computed by the model showed near perfect agreement with the actual upstream hydrograph.