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\[\int\limits_{A}{\nabla \cdot \left( {{\Gamma }^{\phi }}h\nabla \phi \right)}\text{d}A=\oint\limits_{S}{{{\Gamma }^{\phi }}h\left( \nabla \phi \cdot \mathbf{n} \right)}\text{d}S=\sum\limits_{f}^{{}}{\bar{\Gamma }_{f}^{\phi }{{{\bar{h}}}_{f}}\Delta {{l}_{f}}{{\left( {{{\hat{n}}}_{i}}{{\nabla }_{i}}\phi \right)}_{f}}}\]

First the momentum equations are rewritten as

$\frac{\partial (h{U}_i)}{\partial t}+\frac{\partial }{\partial x_j}((h{U}_i{U}_j)-{\nu }_{t}h\frac{\partial U_i}{\partial x_j})=-gh\frac{\partial \eta }{\partial x_i}+S_i$

(1)

where $S_i$ includes all other terms. The equation is then integrated over the a control volume as

$\frac{\partial }{\partial t}\underset{A}{\int }h{U}_{i}dA+{\oint }_F\frac{\partial }{\partial x_j}[(h{U}_i{U}_j)-{\nu }_{t}h\frac{\partial U_i}{\partial x_j}]dF=-gh{\oint }_F\frac{\partial \eta }{\partial x_i}dF+\underset{A}{\int }{S}_{i}dA$

(2)

The resulting di

$U_{i,P}^{n+1}=\frac{1}{a_{i,P}}(\sum _{k=1}{a}_{i,k}{U}_{i,k}^{n+1}+S_i)-\frac{h_P}{a_{i,P}}\sum _{k=1}{n}_{ik}\mathrm{\Delta }{s}_k{p}_k^{n+1}$

The continuity equation is discretized as

$h^{n+1}-\mathbf{𝐒}$

where the subscript $k$ indicates the cell face, $p=g\eta$ with $\eta$ being the water surface elevation, $n_{ik}$ is equal to the dot product of the velocity unit vector and the cell face unit vector.

The coefficient $a_{i,P}$ is equal to $a_{i,P}=\sum a_{i,k}+a_P^0$

The continuity equation is discretized as $p_P^{n+1}=p_P^n-g\frac{\mathrm{\Delta }t}{\mathrm{\Delta }{A}_P}\sum _{k=1}{n}_k{F}_k^{n+1}$

where $n_k$ is the dot product of the cell face unit vector and

The depth-averaged 2-D continuity and momentum equations are given by

$\frac{\partial h}{\partial t}+\frac{\partial (h{U}_j)}{\partial x_j}=S$

(1)

for $j=1,2$