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Title: 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 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 methodmore » reduces numerical oscillations and instabilities without requiring ad hoc smoothing algorithms.« less

Authors:
 [1]; ORCiD logo [1];  [2]
  1. Univ. of Texas, Austin, TX (United States)
  2. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1665967
Grant/Contract Number:  
AC05-00OR22725
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Hydrology and Earth System Sciences (Online)
Additional Journal Information:
Journal Volume: 24; Journal Issue: 8; Journal ID: ISSN 1607-7938
Publisher:
European Geosciences Union (EGU)
Country of Publication:
United States
Language:
English
Subject:
54 ENVIRONMENTAL SCIENCES

Citation Formats

Yu, Cheng-Wei, Hodges, Ben R., and Liu, Frank. A new form of the Saint-Venant equations for variable topography. United States: N. p., 2020. Web. doi:10.5194/hess-24-4001-2020.
Yu, Cheng-Wei, Hodges, Ben R., & Liu, Frank. A new form of the Saint-Venant equations for variable topography. United States. doi:10.5194/hess-24-4001-2020.
Yu, Cheng-Wei, Hodges, Ben R., and Liu, Frank. Tue . "A new form of the Saint-Venant equations for variable topography". United States. doi:10.5194/hess-24-4001-2020. https://www.osti.gov/servlets/purl/1665967.
@article{osti_1665967,
title = {A new form of the Saint-Venant equations for variable topography},
author = {Yu, Cheng-Wei and Hodges, Ben R. and Liu, Frank},
abstractNote = {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.},
doi = {10.5194/hess-24-4001-2020},
journal = {Hydrology and Earth System Sciences (Online)},
issn = {1607-7938},
number = 8,
volume = 24,
place = {United States},
year = {2020},
month = {8}
}

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