A staggered conservative scheme for every Froude number in rapidly varied shallow water flows

TLDR

Rapidly varied flows, such as hydraulic jumps and bores, cause sudden transitions and steep bathymetric gradients that challenge grid resolution and hydrostatic assumptions, making conservation properties essential. The study proposes a staggered‑grid, implicit integration scheme that accurately handles rapidly varied shallow‑water flows, including expansions and contractions. The scheme uses staggered grids and implicit integration, applying a momentum‑principle‑consistent approximation for expansions and a Bernoulli‑equation‑consistent approximation for contractions. The method converges to the shallow‑water solution under smooth conditions and proves highly efficient for large‑scale inundation simulations. © 2003 John Wiley & Sons, Ltd.

Abstract

Abstract This paper proposes a numerical technique that in essence is based upon the classical staggered grids and implicit numerical integration schemes, but that can be applied to problems that include rapidly varied flows as well. Rapidly varied flows occur, for instance, in hydraulic jumps and bores. Inundation of dry land implies sudden flow transitions due to obstacles such as road banks. Near such transitions the grid resolution is often low compared to the gradients of the bathymetry. In combination with the local invalidity of the hydrostatic pressure assumption, conservation properties become crucial. The scheme described here, combines the efficiency of staggered grids with conservation properties so as to ensure accurate results for rapidly varied flows, as well as in expansions as in contractions. In flow expansions, a numerical approximation is applied that is consistent with the momentum principle. In flow contractions, a numerical approximation is applied that is consistent with the Bernoulli equation. Both approximations are consistent with the shallow water equations, so under sufficiently smooth conditions they converge to the same solution. The resulting method is very efficient for the simulation of large‐scale inundations. Copyright © 2003 John Wiley & Sons, Ltd.