Application of the implicit method to surges in open channels

A technique is proposed by which continuous numerical computation of flows with surges in open channels can be handled with an implicit formulation. The one‐dimensional gradually varied flow equations of continuity and momentum are written in finite difference form for the point (I + ½, J + θ) in the x, t plane, where I is the distance counter, J is the time counter, and θ is a weighting parameter that is varied from 0.5 to 1.0. A computer program was written to solve the problem in which the discharge hydrograph is given at the upstream end of a channel and a stage‐discharge relationship is specified at the downstream end. This program was applied to different flow situations in laboratory channels and natural rivers. It was found that θ = 0.5 produced results with a steep wave front but with considerable numerical instability. The use of θ = 1.0 gave results with considerable diffusion of the wave front. The use of θ = 0.60 produced results that had a steep wave front and only a minor degree of numerical instability. This value of θ resulted in good agreement between calculated and measured hydrographs at the downstream end of the channel in most applications. However, the optimum value of θ depends on channel friction and the steepness of the wave front.