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$\underset{A}{\int }\mathrm{\nabla }\cdot \left({\mathrm{\Gamma }}^{\varphi }h\mathrm{\nabla }\varphi \right)\text{d}A=\underset{S}{\oint }{\mathrm{\Gamma }}^{\varphi }h(\mathrm{\nabla }\varphi \cdot \mathbf{𝐧})\text{d}S=\sum _f^{}{\overline{\mathrm{\Gamma }}}_f^{\varphi }{\overline{h}}_f\mathrm{\Delta }{l}_f{\left({\hat{n}}_i{\mathrm{\nabla }}_i\varphi \right)}_f$

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 $j=1,2$