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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

${\frac {\partial (hU_{i})}{\partial t}}+{\frac {\partial }{\partial x_{j}}}{\biggl (}(hU_{i}U_{j})-\nu _{t}h{\frac {\partial U_{i}}{\partial x_{j}}}{\biggr )}=-gh{\frac {\partial \eta }{\partial x_{i}}}+S_{i}$

(1)

where $S_{i}$ includes all other terms. The equation is then integrated over the a control volume as

${\frac {\partial }{\partial t}}\int _{A}hU_{i}dA+\oint _{F}{\frac {\partial }{\partial x_{j}}}{\biggl [}(hU_{i}U_{j})-\nu _{t}h{\frac {\partial U_{i}}{\partial x_{j}}}{\biggr ]}dF=-gh\oint _{F}{\frac {\partial \eta }{\partial x_{i}}}dF+\int _{A}S_{i}dA$

(2)

The resulting di

$U_{i,P}^{n+1}={\frac {1}{a_{i,P}}}{\biggl (}\sum _{k=1}a_{i,k}U_{i,k}^{n+1}+S_{i}{\biggr )}-{\frac {h_{P}}{a_{i,P}}}\sum _{k=1}n_{ik}\Delta s_{k}p_{k}^{n+1}$

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 $p_{P}^{n+1}=p_{P}^{n}-g{\frac {\Delta t}{\Delta A_{P}}}\sum _{k=1}n_{k}F_{k}^{n+1}$

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

${\frac {\partial h}{\partial t}}+{\frac {\partial (hU_{j})}{\partial x_{j}}}=S$

(1)

for $j=1,2$