## 2 Lagrangian and Eulerian Methods
### Non-Hydrostatic Discontinuous/Continuous Galerkin Model
The effects of the turbulence, Coriolis force, atmospheric pressure and baroclinic pressure gradient, are traditionally ignored in the study of non-hydrostatic wave propagation. However, simulation with these effects is possible; see, e.g., [PERSON] and [PERSON] (2018). The average vertical velocity, \(W\), is \((w_{\xi}+w_{b})/2\), where \(w_{\xi}\) is the vertical velocity at the surface and \(w_{b}\) is the vertical velocity at the bottom. The kinematic boundary conditions of the free surface and the bottom are:  
\[w_{\xi}=\frac{\partial\xi}{\partial t}+u_{\xi}\frac{\partial\xi}{\partial x}+ v_{\xi}\frac{\partial\xi}{\partial y} \tag{12}\]  
\[w_{b}=-u_{b}\frac{\partial h}{\partial x}-v_{b}\frac{\partial h}{\partial y} \tag{13}\]  
with \(u_{\xi},v_{\xi},u_{b}\)_and_\(v_{b}\) being the components in x and y of the velocities next to the free surface and the bottom, respectively. Due to the linearization of the vertical momentum equation, it is well recognized that the depth-integrated non-hydrostatic models are only appropriate for weakly nonlinear cases ([PERSON], 2018). The solution begins by solving the horizontal momentum Equations (8) and (9), without the non-hydrostatic pressure terms, by means of a discontinuous Galerkin method. These horizontal momentum Equations, coupled with the mass conservation Equation (10), can be written in the conservative form of Equation (14). \[\frac{\partial\mathbf{U}}{\partial t}+\
abla.\mathbf{F}(\mathbf{U})=\frac{\partial\mathbf{U}} {\partial t}+\frac{\partial\mathbf{E}(\mathbf{U})}{\partial x}+\frac{\partial\mathbf{G}( \mathbf{U})}{\partial y}=\mathbf{S}(\mathbf{U}) \tag{14}\]  
The vectors of conserved variables \(\mathbf{U}\), source vector \(\mathbf{S}\), and flux vectors \(\mathbf{F}(\mathbf{U})\) are given in Equations (15) to (18). \[\mathbf{U}=\left\{ \begin{array}{c}q_{x}\\ q_{y}\\ H\end{array} \right\} \tag{15}\]  
\[\mathbf{S}=\left\{ \begin{array}{c}-gH\frac{\partial\xi}{\partial x}-\frac{gn^{2} l\sqrt{U^{2}+V^{2}}}{H^{2}/3}\\ -gH\frac{\partial\xi}{\partial y}-\frac{gn^{2}V^{2}V^{2}U^{2}+V^{2}}{H^{1/3}} \\ 0\end{array} \right\} \tag{16}\]  
\[\mathbf{F}=(\mathbf{E}(\mathbf{U}),\mathbf{G}(\mathbf{U})) \tag{17}\]  
\[\mathbf{E}(\mathbf{U})=\left\{ \begin{array}{c}q_{x}^{2}/H\\ q_{x}q_{y}/H\\ q_{x}\end{array} \right\},\mathbf{G}(\mathbf{U})=\left\{ \begin{array}{c}q_{x}q_{y}/H \\ q_{y}^{2}/H\\ q_{y}\end{array} \right\} \tag{18}\]  
In these Equations, \(q_{x}\) and \(q_{y}\) are discharges per unit width in the \(x\) and \(y\) directions and are equal to \(HU\) and \(HV\), respectively. The discontinuous Galerkin formulation of the governing Equation (14) is obtained by multiplying it with a shape function \(\varphi\) and integrate over an element, \(\Omega_{e}\). The flux term \(\mathbf{F}\) is integrated using Gauss theorem, resulting in Equation (19). In this equation, \(\mathbf{n}=\left(n_{x},n_{y}\right)\) is the outward unit normal vector at an element boundary \(\Gamma_{e}\). \[\int_{\Omega_{e}}\varphi\frac{\partial\mathbf{U}}{\partial t}d\Omega+\int_{ \Gamma_{e}}\varphi\mathbf{F}.\mathbf{n}\ d\Gamma-\int_{\Omega_{e}}F.\
abla\varphi\ d \Omega=\int_{\Omega_{e}}\varphi\ \mathbf{S}\ d\Omega \tag{19}\]In the discontinuous Galerkin method, the variable vector \(\mathbf{U}\) is approximated over a quadrilateral element by:  
\[\mathbf{U}(x,\;y)\approx\sum_{j=1}^{4}U_{j}\phi_{j}(x,\;y) \tag{20}\]  
where \(U_{j}\) are nodal values of the variables and \(\phi_{j}(x,\;y)\) are the bilinear approximation functions of the solution variables or shape functions. Since in a discontinuous Galerkin context the discontinuities of variables at the element boundaries are allowed, the intercell flux is taken as being dependent on the values \(\mathbf{U}\) in each of the two adjacent elements.