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${\frac {\partial (hU_{i})}{\partial t}}+{\frac {\partial }{\partial x_{j}}}{\biggl (}(hU_{i}U_{j})-\nu _{t}h{\frac {\partial U_{i}}{\partial x_{j}}}{\biggr )}dS-\epsilon _{ij3}f_{c}U_{j}h=-gh{\frac {\partial \eta }{\partial x_{i}}}-{\frac {h}{\rho _{0}}}{\frac {\partial p_{a}}{\partial x_{i}}}+{\frac {\tau _{i}}{\rho }}$

(2)

The discretized momentum equations are ${\frac {\partial }{\partial t}}\int _{A}hU_{i}dA+\oint _{A}{\frac {\partial (hU_{i}U_{j})}{\partial x_{j}}}dS-{\frac {h}{\rho _{0}}}{\frac {\partial p_{a}}{\partial x_{i}}}+{\frac {\partial }{\partial x_{j}}}{\biggl (}\nu _{t}h{\frac {\partial U_{i}}{\partial x_{j}}}{\biggr )}-\epsilon _{ij3}f_{c}U_{j}h=-gh{\frac {\partial \eta }{\partial x_{i}}}+{\frac {\tau _{i}}{\rho }}$

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

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 $j=1,2$