An analytical solution to the kinematic wave approximation for unsteady flow routing is presented. The model allows time‐dependent lateral inflow with piecewise spatial uniformity and can be applied to complex kinematic cascades. Kinematic shocks are considered as manifestations of higher‐order effects such as rnonoclinal flood waves, bores, etc. Within the context of kinematic approximation therefore we retain their dynamic effects by routing the discontinuities as they appear. Certain simplifying assumptions are made which permit closed form solutions and an efficient numerical algorithm, based on the method of characteristics, is employed. The resulting model, called an approximate shock‐fitting scheme, preserves the effect of the shocks without the usual computational complications and compares favorably with an implicit finite difference solution. The efficiency and accuracy of the new method are illustrated by computing a variety of unsteady flows, ranging from simple cascades to complex natural water
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