Abstract. 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.
Relative cohomology and volume forms