A new form of the Saint–Venant equations for variable topography

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 source terms and eliminate potential numerical oscillations. It 5 is shown that a simple algebraic transformation of channel geometry provides a smooth reference slope while preserving the correct cross-sectional 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 SPRNT model in a series of artificial test cases and simulation of a small urban creek. Validation comparisons are made with analytical 10 solutions and the HEC-RAS model. The new method reduces numerical oscillations and instabilities without requiring ad hoc smoothing algorithms.