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The discretized momentum equations are

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Double subscripts: use braces to clarify"): {\displaystyle U_{i,P}^{n+1}={\frac {1}{a_{i,P}}}{\biggl (}\sum _{k=1}^{N}_{face}a_{i,k}U_{i,k}^{n+1}+S_{i}{\biggr )}-{\frac {h_{P}}{a_{i,P}}}\sum _{k=1}^{N}_{face}n_{i}n_{k}\Delta s_{k}p_{k}^{n+1}}

where the subscript $k$ indicates the cell face, $p=g\eta$ with $\eta$ being the water surface elevation, $n_{i}$ is the unit vector in the $i$ direction, and $n_{k}$ is the unit vector normal to the cell face.

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

The continuity equation is discretized as

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 $j=1,2$

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

(2)