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The discretized momentum equations are Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial h}{\partial t} \int_{A_p} U_i + \oint_{F} \frac{\partial (h U_i U_j )}{\partial x_j} - \epsilon_{ij3} f_c U_j h = - g h \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} }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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

where the subscript Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k} indicates the cell face, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p = g \eta} with Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \eta} being the water surface elevation, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_{ik}} is equal to the dot product of the velocity unit vector and the cell face unit vector.

The coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_{i,P}} is equal to Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_{i,P} = \sum a_{i,k} + a_P^0 }

The continuity equation is discretized as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_P^{n+1} = p_P^n - g \frac{\Delta t}{\Delta A_P} \sum_{k=1} n_k F_k^{n+1}}

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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

(1)

for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j=1,2 }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial ( h U_i ) }{\partial t} + \frac{\partial (h U_i U_j )}{\partial x_j} - \epsilon_{ij3} f_c U_j h = - g h \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)

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