## 2 Lagrangian and Eulerian Methods
### Non-Hydrostatic Discontinuous/Continuous Galerkin Model
Thus, the normal flux \(\mathbf{F}.\mathbf{n}\) is not exclusively defined and is substituted by a numerical flux \(\widetilde{\mathbf{F}}(\mathbf{U}_{L},\mathbf{U}_{R})\) where \(\mathbf{U}_{L}\) and \(\mathbf{U}_{R}\) are the variables at the left (internal) and right (external) sides of the element boundary in the counterclockwise direction, respectively. Consequently, the second integral in Equation (19) is expressed as:  
\[\int_{\mathbf{\Gamma}_{e}}\varphi\mathbf{F}.\mathbf{n}\;d\Gamma=\int_{\mathbf{\Gamma}_{e}} \varphi\widetilde{\mathbf{F}}d \tag{21}\]  
In this analysis, \(\widetilde{\mathbf{F}}\) is chosen to be the common Harten-Lax-van Leer (HLL) numerical flux [83]. The first step of the solution process ends with the intermediate estimation of the conserved variables \(q_{x}\) and \(q_{y}\) from Equation (17): \(\widetilde{q}_{x}^{n+1}\)_and_\(\widetilde{q}_{y}^{n+1}\). In the second step of the solution, a Poisson equation is constructed that is then solved by a continuous Galerkin method to obtain the non-hydrostatic pressures \(q_{b}^{n+1}\). A time approximation of the vertical moment Equation in (11) is:  
\[w_{\xi}^{n+1}=w_{\xi}^{n}-w_{b}^{n+1}+w_{b}^{n}+\frac{2\Delta tq_{b}^{n+1}}{ \rho\bar{H}^{n}} \tag{22}\]  
The vertical velocity at the bottom is valued from the kinematic boundary condition (16), estimated as:  
\[w_{b}^{n+1}=-U^{n}\frac{\partial h}{\partial x}-V^{n}\frac{\partial h}{ \partial y} \tag{23}\]  
\[w_{b}^{n}=-U^{n-1}\frac{\partial h}{\partial x}-V^{n-1}\frac{\partial h}{ \partial y} \tag{24}\]  
From the remaining part of the horizontal momentum, Equations (8) and (9) including the non-hydrostatic pressure terms:  
\[\frac{\partial(HU)}{\partial t}=-\frac{H}{2\rho}\frac{\partial q_{b}}{ \partial x}-\frac{q_{b}}{2\rho}\frac{\partial(\xi-h)}{\partial x} \tag{25}\]  
\[\frac{\partial(HV)}{\partial t}=-\frac{H}{2\rho}\frac{\partial q_{b}}{ \partial y}-\frac{q_{b}}{2\rho}\frac{\partial(\xi-h)}{\partial y} \tag{26}\]  
The final horizontal velocities, influenced by the non-hydrostatic pressure, can be time approximated as  
\[U_{c}^{n+1}=\widetilde{U}^{\;n+1}-\frac{\Delta t}{2\rho}\frac{\partial q_{b} ^{n+1}}{\partial x}-\frac{\Delta t}{2\rho}\frac{q_{b}^{n+1}}{H_{c}^{n}}\frac{ \partial}{\partial x}(\xi_{c}^{n}-h) \tag{27}\]  
\[V_{c}^{n+1}=\widetilde{V}^{\;n+1}-\frac{\Delta t}{2\rho}\frac{\partial q_{b} ^{n+1}}{\partial y}-\frac{\Delta t}{2\rho}\frac{q_{b}^{n+1}}{H_{c}^{n}}\frac{ \partial}{\partial y}(\xi_{c}^{n}-h) \tag{28}\]  
In the continuous Galerkin solution of the non-hydrostatic pressures, the continuous horizontal velocities \(U_{c}^{n+1}\) and \(V_{c}^{n+1}\), continuous water level \(\xi_{c}^{n}\) and flow depth \(H_{c}^{n}\), vertical velocities \(w_{\xi}\) and \(w_{b}\), and non-hydrostatic pressures \(q_{b}^{n+1}\) are approximated with nodal variables, which have the same value in all the adjoining elements. The continuous intermediate velocities \(\widetilde{U}\ ^{n+1}\) and \(\widetilde{V}\ ^{n+1}\) are obtained from the intermediate evaluation of the discontinuous discharges \(\widetilde{q}_{x}^{n+1}\)_and_\(\widetilde{q}_{y}^{n+1}\) integrating over the entire domain \(\Omega\). \[\mathbf{U}(x,\;y)\approx\sum_{j=1}^{4}U_{j}\phi_{j}(x,\;y) \tag{20}\]0  
\[\mathbf{U}(x,\;y)\approx\sum_{j=1}^{4}U_{j}\phi_{j}(x,\;y) \tag{20}\]1  
where the discontinuous intermediate velocities \(\widetilde{U}_{dc}^{n+1}\) and \(\widetilde{V}_{dc}^{n+1}\) are approximated in an element using the values of \(\frac{\widetilde{q}_{x}^{n+1}}{H^{n}}\) and \(\frac{\widetilde{q}_{y}^{n+1}}{H^{n}}\), respectively, evaluated at the nodes. Similarly, the continuous water level elevations \(\frac{\widetilde{q}_{c}^{n}}{c}\) are obtained from:  
\[\mathbf{U}(x,\;y)\approx\sum_{j=1}^{4}U_{j}\phi_{j}(x,\;y) \tag{20}\]2  
where \(\xi^{n}=H^{n}-h\). Finally, the continuous flow depth, \(H_{c}^{n}\), is \(H_{c}^{n}=\xi_{c}^{n}+h\).