Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage

New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.

Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage

New finite-volume forms of the Saint-Venant equations for one-dimensional (1D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and serve to transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. The derivation introduces an analytical approximation of the free surface across a finite volume element (e.g. linear, parabolic) as well as an analytical approximation of the bottom topography. Integration of the product of these provides an approximation of a piezometric pressure gradient term that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, water surface elevations, and the channel bottom elevation (but without using any volume-averaged bottom slope). The new conservative form should be more tractable for large-scale simulations of river networks and urban drainage systems than the traditional conservative form of the Saint-Venant equations where it is difficult to maintain a well-balanced discretization for highly-variable topography.