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Journal of Applied Nonlinear Dynamics 12(3) (2023) 547--570 | DOI:10.5890/JAND.2023.09.009

Chinedu Nwaigwe$^1$, Andreas S. Dedner$^2$

$^1$ Department of Mathematics, Rivers State University, Port Harcourt, Nigeria

$^2$ Warwick Mathematics Institute, University of Warwick, United Kingdom

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Abstract

In this paper, we propose the subcell method to couple channel and flood flows. We adopt a 1D Saint Venant channel model with coupling terms and the 2D shallow water flood model. The channel flow is coupled to the flood through the discrete 1D coupling term which we derived in a closed form; while the flood is coupled to the channel flow through the 2D numerical fluxes. Since 1D channel models ignore the evolution of channel lateral discharges which are needed to compute the 2D numerical fluxes at flood/channel interfaces, the problem of recovering channel lateral discharges is a crucial one. To this end, we propose a technique that splits channel cells into two sub-cells and adopt an ad-hoc model based on the y-discharge equation in the 2D shallow water equations. Then, motivated by the hydrostatic reconstruction scheme, the subcell hydrostatic reconstruction scheme is formulated, for the first time, and used to compute the channel lateral discharges. This constitutes the novelty of this work. Also, deriving the 1D discrete coupling term in closed form is another novelty. This approach can be easily implemented without requiring any change to the existing channel or flood solver. We prove that the proposed method is well-balanced and satisfies a \emph{no-numerical flooding} property, and present some numerical test cases on constant-width channels and rectangular floodplains to demonstrate the accuracy and performance of the method. Our results show that the method computes results with good accuracy, yet performs well. We therefore conclude that including a model for evolving lateral discharges within the channel during a flooding event, leads to a significant improvement in the accuracy of the scheme.

Acknowledgments

We are grateful to the anonymous Reviewers whose suggestions greatly improved the work.

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