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In this paper, we propose a novel approach for coupling 2D/1D shallow water flow models. Efficiently coupling these models is vital for simulating the flow and flooding of open channels. Currently, existing methods couple the models either at the channel lateral boundaries (lateral methods) or at the location, along the channel flow direction, where the two sub-domains intersect (frontal methods). We classify these methods as horizontal methods since the coupling points are on the horizontal plane. The limitations of these methods include their inability to recover the 2D channel flow structure during flooding without losing efficiency, and their inability to switch between lateral and frontal methods, based on the problem. Here, we propose a new paradigm which is to think of the channel flow as a two layer flow. This leads to the vertical coupling method (VCM) which is able to recover 2D channel flow structure without losing efficiency, switch types depending on the flow and is a superset of some existing methods. The VCM is based on a user chosen elevation above which the user considers the channel to be full. From this elevation, all other flow quantities are defined including the two layers. The flows in the lower and upper layers are assumed to be 1D and 2D respectively, and the appropriate flow models with exchange terms are derived. The 2D shallow water flow model is retained for the floodplains. Finite volume methods (FVM) are formulated for the flood model; the FVM with the operator splitting approach are also formulated to solve and couple the two layer channel flow models. We show that the resulting method (i) is well-balanced (ii) preserves the "no-numerical" flooding property (iii) preserves conservation properties and (iv) adapts to the flow situation., Comment: 49 pages, 26 figures. arXiv admin note: text overlap with arXiv:1701.02577

One dimensional (1D) simulations of the flow and flooding of open channels are known to be inaccurate as the flow is multi-dimensional in nature, especially at the flooded regions. However, multi-dimensional simulations, even in two dimensions (2D), are computationally expensive, hence the problem of efficiently coupling 2D and 1D simulations for the flow and flooding of open channels has been the subject of much research and is investigated in this paper. We adopt a 1D model with coupling term for the channel flow and the 2D shallow water flow model for the floodplain. The 1D model with coupling term is derived by integrating the 3D Free Surface Euler equations but without imposing any restriction on the channel width variations. %leading to a new coupling term. Finite volume methods are formulated for both the 2D and 1D models including a discrete coupling term in closed form. Coupling is achieved through the discrete coupling term in the 1D model and the lateral numerical fluxes in the 2D model. Since the lateral discharge in the channel cannot be guaranteed to be zero during flooding, we aim to recover the lateral variation by computing two lateral discharges over each cross section and propose to use an ad-hoc model based on the $y$-discharge equation in the 2D model for this purpose. We then propose the numerical scheme for this ad-hoc model following the hydrostatic reconstruction philosophy. Then, we show that the resulting method, named Horizontal Coupling Method (HCM), is well-balanced; we introduce the no-numerical flooding property and also show that the method satisfies the property. Three numerical test cases are used to verify the performance of the method. The results show that the method performs well in both accuracy and efficiency and also approximates the channel lateral discharges with very good accuracy and little computational overhead., Comment: 32 pages, 23 figures