  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.
