[Paper Review] A well-balanced reconstruction for wetting/drying fronts

Andreas Bollermann, Guoxian Chen+2 more|arXiv (Cornell University)|Dec 11, 2014

Computational Fluid Dynamics and Aerodynamics31 references3 citations

TL;DR

This paper presents a well-balanced, positivity-preserving finite volume scheme for the shallow water equations using a continuous, piecewise linear reconstruction of bottom topography. The key innovation is a novel reconstruction strategy for wet/dry cells that ensures exact preservation of both 'lake at rest' and 'dry lake' steady states, with positivity maintained via flux limiting at local draining times, enabling robust simulation of wetting and drying fronts without time step restrictions.

ABSTRACT

In this paper, we construct a well-balanced, positivity preserving finite volume scheme for the shallow water equations based on a continuous, piecewise linear discretization of the bottom topography. The main new technique is a special reconstruction of the flow variables in wet-dry cells, which is presented in this paper for the one dimensional case. We realize the new reconstruction in the framework of the second-order semi-discrete central-upwind scheme from (A. Kurganov and G. Petrova, Commun. Math. Sci., 2007). The positivity of the computed water height is ensured following (A. Bollermann, S. Noelle and M. Lukáčová-Medviďová, Commun. Comput. Phys., 2010): The outgoing fluxes are limited in case of draining cells.

Motivation & Objective

• To develop a well-balanced finite volume scheme that exactly preserves both 'lake at rest' and 'dry lake' steady states in shallow water simulations.
• To ensure positivity of water height in wet/dry cells, especially when numerical oscillations threaten to drive depth negative.
• To maintain second-order accuracy and robustness in the presence of dry areas and shocks impinging on dry zones.
• To enable stable and accurate simulation of wetting and drying fronts without reducing the global time step.
• To extend the central-upwind scheme to handle continuous, piecewise linear bottom topography while preserving well-balancing and positivity.

Proposed method

• A new reconstruction technique is proposed for flow variables in wet/dry cells, specifically designed to preserve both 'lake at rest' and 'dry lake' steady states.
• The method uses a continuous, piecewise linear discretization of the bottom topography to improve numerical stability and accuracy.
• Positivity of water height is enforced by limiting outgoing fluxes when the local draining time is shorter than the global time step.
• The scheme is built within the framework of the second-order semi-discrete central-upwind finite volume method from Kurganov & Petrova (2007).
• The reconstruction ensures that the numerical fluxes balance the source terms exactly at equilibrium states, including mixed wet/dry configurations.
• A partially implicit treatment is applied to the Manning friction term to handle its singularity at wet/dry fronts without destabilizing the scheme.

Experimental results

Research questions

• RQ1 Can a well-balanced, positivity-preserving scheme be constructed for the shallow water equations using continuous, piecewise linear bottom topography reconstruction?
• RQ2 How can wet/dry fronts be reconstructed to preserve both 'lake at rest' and 'dry lake' steady states simultaneously?
• RQ3 Can positivity of water height be ensured without imposing a restrictive time step constraint?
• RQ4 How does the scheme perform in the presence of shocks impinging on dry areas or strong friction terms?
• RQ5 Does the proposed reconstruction maintain second-order accuracy and well-balancing in non-equilibrium, transient flows?

Key findings

• The proposed scheme exactly preserves 'lake at rest' and 'dry lake' steady states, even in mixed wet/dry domains, due to the specially designed reconstruction.
• The scheme maintains second-order accuracy in space and achieves experimental order of convergence in all tested numerical examples.
• Positivity of water height is preserved across wet/dry fronts by flux limiting at the local draining time, avoiding the need for time step reduction.
• The method successfully simulates a laboratory dam-break over a triangular hump with Manning friction, matching measured water depths at seven gauge points with high accuracy.
• The scheme is robust for shocks running into dry areas and handles the singular nature of the Manning friction term effectively.
• A case study shows that a previous version of the scheme fails to converge to equilibrium, demonstrating the necessity of the new reconstruction.