A COMPARATIVE ANALYSIS OF A SPACE-TIME CE/SE FINITE VOLUME SCHEME FOR POROSITY-DEPENDENT SHALLOW WATER EQUATIONS

Abstract

Simulating shallow water flows over a porous or obstructed terrain accurately is an important problem in hydraulic engineering, geophysical fluid dynamics and environmental science. This paper investigates the one-dimensional shallow water equations (SWEs) with porosity, which is a class of shallow water equations that extends the classical Saint-Venant system of equations to include the volume of fluid space occupied by porous or partially obstructed flow domains. The introduction of porosity adds to the flux functions, and to the source terms of the governing hyperbolic system, creating additional mathematical complexity that can give rise to challenging numerical discretization issues, but that retains the face of the original equations. This space-time Conservation Element and Solution Element (CE/SE) method is a finite volume-based scheme that unifies the temporal and spatial dimension in a geometric manner to address these challenges, is rigorously extended and implemented for the SWEs that depend on porosity. The scheme is consistent with the conservation of flux on both spatial and temporal control surfaces using the Gauss divergence theorem and does not need an artificial viscosity, and it naturally prevents numerical oscillations around discontinuities by using a weighted-average limiter. It is shown that the proposed scheme is positive for water level and is well-balanced, i.e., it precisely reproduces the Lake-at-Rest steady states. The results of the CE/SE scheme show improved resolution of stationary contact discontinuities at porosity jumps and sharp shock profiles with the comparable accuracy of the smooth flow regions, while both schemes perform very well in smooth flow regions. The results demonstrate that the CE/SE method is a strong and precise method for the shallow water modeling which is efficient computationally and can be applied to modeling with porosity