[Paper Review] A central-upwind scheme for open water flow in a wet/dry multiply-connected channel network

Sergii Kivva, Mark Zheleznyak+2 more|arXiv (Cornell University)|Aug 17, 2017

Computational Fluid Dynamics and Aerodynamics54 references4 citations

TL;DR

This paper presents a novel central-upwind finite-volume scheme for 1D shallow water flow in complex, multiply-connected channel networks with arbitrary geometry and variable topography. It introduces a positivity-preserving, well-balanced reconstruction of water surface elevation in partially flooded cells without correction, combined with exact source term integration, enabling robust simulation of subcritical and supercritical flows with wetting and drying across junctions.

ABSTRACT

Our goal was to develop a robust algorithm for numerical simulation of one-dimensional shallow-water flow in a complex multiply-connected channel network with arbitrary geometry and variable topography. We apply a central-upwind scheme with a novel reconstruction of the open water surface in partially flooded cells that does not require additional correction. The proposed reconstruction and an exact integration of source terms for momentum conservation equation provide positivity preserving and well-balanced features of the scheme for various wet-dry states. We use two models based on continuity equation and mass and momentum conservation equations integrated for a control volume around the channel junction to its treatment. These junction models permit to simulate a subcritical and supercritical flow in a channel network. Numerous numerical experiments demonstrate the robustness of the proposed numerical algorithm and a good agreement of numerical results with exact solutions, experimental data, and results of the previous numerical studies. The proposed new specialized test on inundation and drying of initially dry channel network shows the merits of the new numerical algorithm to simulate the subcritical/supercritical open water flows in the network.

Motivation & Objective

• To develop a robust numerical scheme for simulating 1D shallow water flow in multiply-connected channel networks with irregular geometry and variable topography.
• To address the limitations of existing schemes like Preissmann, which fail to preserve positivity in drying channels.
• To ensure well-balancing and positivity preservation during wetting and drying processes without artificial corrections.
• To provide a physically consistent treatment of channel junctions that supports both subcritical and supercritical flow regimes.
• To validate the scheme against exact solutions, experimental data, and prior numerical studies, including a new test case for inundation and drying.

Proposed method

• A second-order accurate central-upwind finite-volume scheme is applied to the 1D shallow water equations in conservative form.
• A novel reconstruction of water surface elevation is proposed for partially flooded cells, generalizing prior work and eliminating the need for additional correction steps.
• Exact integration of source terms (due to bed slope and variable width) is performed using linear approximations of water and bed elevations at cell faces.
• Two junction models are developed: one based on continuity and energy balance for subcritical flow, and another using mass and momentum conservation for supercritical flow.
• The local draining time approach limits outflow flux from drying cells to preserve positivity without reducing the CFL time step.
• Implicit treatment of only part of the friction slope (following Chertok et al.) maintains stability without additional time step restrictions.

Experimental results

Research questions

• RQ1 Can a central-upwind scheme be adapted to maintain positivity and well-balancing in 1D shallow water flow simulations across complex, multiply-connected channel networks with arbitrary geometry?
• RQ2 How can water surface elevation be reconstructed in partially flooded cells without requiring additional correction steps or introducing numerical artifacts?
• RQ3 What is the impact of exact source term integration on the well-balancing property for various wet-dry states and steady flows?
• RQ4 How do the proposed junction models compare to standard assumptions (e.g., equal water surface elevation) in handling subcritical and supercritical flows?
• RQ5 Can the scheme robustly simulate dynamic inundation and drying processes in a realistic channel network, including wave propagation and flow reversal?

Key findings

• The proposed reconstruction of water surface elevation in partially flooded cells successfully maintains positivity without additional correction, improving robustness in drying scenarios.
• Exact integration of source terms ensures the scheme is well-balanced for all steady states, including those at rest with wet/dry transitions.
• Numerical experiments show excellent agreement with exact solutions, experimental data, and previous numerical studies, confirming accuracy and convergence.
• The new test case for inundation and drying demonstrates the scheme’s ability to simulate supercritical and subcritical flows across a complex network with stable, physically consistent results.
• The junction models correctly handle both subcritical and supercritical flow regimes, with numerical results showing non-zero net discharge at junctions—contradicting the common assumption of discharge balance.
• The scheme maintains stability and positivity even under high CFL numbers (e.g., 0.9), with no need to reduce time steps due to drying cells.