Difference Solutions of the Shallow-Water Equation

by James A. Liggett,
David A. Woolhiser,

Serial Information: Journal of the Engineering Mechanics Division, 1967, Vol. 93, Issue 2, Pg. 39-72

Document Type: Journal Paper


The quasi-linear partial differential equations known as the shallow-water equations describe the flow of water in open channels or over sloping planes (overland flow). Because no analytic solution exists for these equations, finite-difference methods must be used to obtain solutions. Formulation of finite-difference schemes involves consideration of the convergence of the finite-difference solutions to the true solution of the equation. A theoretical examination of the approximation of several difference operators, both implicit and explicit, is presented and the stability of these schemes is examined empirically for flow over a plane with critical depth downstream boundary condition and a zero inflow upstream boundary condition. A finite-difference scheme based on the method of characteristics was found to be satisfactory in many cases. Explicit methods were found to be not suitable for this problem except in some special cases.

Subject Headings: Finite difference method | Shallow water | Open channel flow | Overland flow | Water flow | Boundary conditions | Open channels | Slopes

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