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One dimensional (1D) simulations of the flow and flooding of open channelsare known to be inaccurate as the flow is multi-dimensional in nature,especially at the flooded regions. However, multi-dimensional simulations, evenin two dimensions (2D), are computationally expensive, hence the problem ofefficiently coupling 2D and 1D simulations for the flow and flooding of openchannels has been the subject of much research and is investigated in thispaper. We adopt a 1D model with coupling term for the channel flow and the 2Dshallow water flow model for the floodplain. The 1D model with coupling term isderived by integrating the 3D Free Surface Euler equations but without imposingany restriction on the channel width variations. %leading to a new couplingterm. Finite volume methods are formulated for both the 2D and 1D modelsincluding a discrete coupling term in closed form. Coupling is achieved throughthe discrete coupling term in the 1D model and the lateral numerical fluxes inthe 2D model. Since the lateral discharge in the channel cannot be guaranteedto be zero during flooding, we aim to recover the lateral variation bycomputing two lateral discharges over each cross section and propose to use anad-hoc model based on the $y$-discharge equation in the 2D model for thispurpose. We then propose the numerical scheme for this ad-hoc model followingthe hydrostatic reconstruction philosophy. Then, we show that the resultingmethod, named Horizontal Coupling Method (HCM), is well-balanced; we introducethe no-numerical flooding property and also show that the method satisfies theproperty. Three numerical test cases are used to verify the performance of themethod. The results show that the method performs well in both accuracy andefficiency and also approximates the channel lateral discharges with very goodaccuracy and little computational overhead.

Formulation, Implementation and Validation of the Horizontal Coupling Method for 1D/2D Shallow Water Flow Models