Abstract
Abstract. The solution stability of river models using the one-dimensional (1D) Saint-Venant equations can be easily undermined when source terms in the discrete equations do not satisfy the Lipschitz smoothness condition for partial differential equations. Although instability issues have been previously noted, they are typically treated as model implementation issues rather than as underlying problems associated with the form of the governing equations. This study proposes a new reference slope form of the Saint-Venant equations to ensure smooth slope source terms and eliminate one source of potential numerical oscillations. It is shown that a simple algebraic transformation of channel geometry provides a smooth reference slope while preserving the correct cross-section flow area and the total Piezometric pressure gradient that drives the flow. The reference slope method ensures the slope source term in the governing equations is Lipschitz continuous while maintaining all the underlying complexity of the real-world geometry. The validity of the mathematical concept is demonstrated with the open-source Simulation Program for River Networks (SPRNT) model in a series of artificial test cases and a simulation of a small urban creek. Validation comparisons are made with analytical solutions and the Hydrologic Engineering Center's River Analysis System (HEC-RAS) model. The new method reduces numerical oscillations and instabilities without requiring ad hoc smoothing algorithms.
Highlights
The Saint-Venant equations (SVEs) for one-dimensional (1D) river modeling are typically presented with pressure forcing terms of either (i) gradients of the water surface elevation or (ii) thalweg bottom slope combined with gradients of the water depth
The results indicate that Simulation Program for River Networks (SPRNT)-reference slope (RS) provides numerical solutions that are nearly identical to Hydrologic Engineering Center’s River Analysis System (HEC-RAS) for the non-smooth geometry test cases
The reference slope (RS) method introduces a new form of the Saint-Venant equations for 1D river flow
Summary
The Saint-Venant equations (SVEs) for one-dimensional (1D) river modeling are typically presented with pressure forcing terms of either (i) gradients of the water surface elevation or (ii) thalweg bottom slope combined with gradients of the water depth. Numerous techniques and special numerical schemes have been previously designed to overcome unwanted numerical oscillations caused by discontinuous geometries and boundary conditions (e.g., Zhou et al, 2001; Liang and Marche, 2009) When a large-scale open-channel model develops oscillations and/or instabilities, practitioners may resort to the traditional approach of removing cross sections or smoothing bathymetry to mitigate oscillatory or unstable solution behavior (Tayfur et al, 1993) Such ad hoc efforts can be effective as they address a major cause of such oscillations and instabilities (discontinuous topography), but they inherently reduce the fidelity of the simulation
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