A new form of the Saint-Venant equations for variable topography
Journal
Hydrology and Earth System Sciences
Journal Volume
24
Journal Issue
8
Pages
4001-4024
Date Issued
2020
Author(s)
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. © Author(s) 2020.
SDGs
Other Subjects
Rivers; Software testing; Topography; Algebraic transformations; Lipschitz continuous; Mathematical concepts; Model implementation; Numerical oscillation; Saint Venant equation; Smoothing algorithms; Smoothness conditions; Numerical methods; algorithm; equation; geometry; hydrological modeling; mathematical concept; numerical method; pressure gradient; slope; smoothing; tomography
Type
journal article