Unsteady, gradually-varied open-channel flow is governed by the Saint-Venant equations — a coupled pair of one-dimensional partial differential equations expressing continuity (conservation of mass) and momentum (conservation of momentum, incorporating channel resistance/friction relationships) for water flow as functions of distance and time. Because these hyperbolic partial differential equations generally admit no closed-form analytical solution, they are solved numerically via finite-difference discretization (using explicit or implicit schemes with forward, backward, or central differences in space and time) to route flood hydrographs through a channel, subject to specified initial and boundary conditions. This belongs to hydraulic/hydrologic engineering, specifically the sub-field of flood/flow routing and computational hydraulics.
Unsteady, gradually-varied open-channel flow is governed by the Saint-Venant equations — a coupled pair of one-dimensional partial differential equations expressing continuity (conservation of mass) …