A second-order well-balanced reconstruction for the shallow flows with wet/dry fronts

초록

In this paper, we develop a two-dimensional, second-order central-upwind scheme based on the finite volume method for the shallow water equations in the presence of wet/dry fronts. The proposed scheme is positivity-preserving and well-balanced, and remains robust for shallow water flows over complex bathymetry and wet-dry interfaces. We extend existing one-dimensional positivity-preserving and well-balanced schemes to two-dimensional spaces, addressing the challenges posed by partially flooded cells with varying bottom gradients in each direction. A novel draining time step for two-dimensional spaces is introduced to ensure the non-negativity of computed water depths across the entire computational domain. The performance of the proposed scheme is validated through several numerical experiments using both analytical solutions and experimental data, demonstrating its accuracy in capturing wet/dry fronts. The results for the lake-at-rest steady-state case confirm that numerical oscillations near the wet/dry fronts are successfully minimized, maintaining the initial state. Moreover, other examples show reduced numerical oscillations during wave run-up and rundown processes, further confirming the performance of the proposed scheme.

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