## 2 Lagrangian and Eulerian Methods
### Non-Hydrostatic Discontinuous/Continuous Galerkin Model  
The Eulerian grid-based method used in this paper is the non-hydrostatic D/C Galerkin, presented in [PERSON] et. al [7]. The model is a two-dimensional depth-integrated non-hydrostatic model with the capacity of simulating wave propagation, breaking and run-up in a finite element framework. Eulerian grid-based depth-integrated models using the Boussinesq-type Equations (BTEs) have been traditionally utilized to simulate wave propagation, but the high order partial derivative terms included in BTEs cause numerical instabilities The shallow water models with non-hydrostatic pressure distribution have shown their capacity for accurate modeling of nonlinear and dispersive waves since their introduction by [PERSON] and [PERSON] [75] and [PERSON] and [PERSON] [76]. The vertical momentum is considered in these models by the introduction of a non-hydrostatic pressure term into the Reynolds-averaged Navier-Stokes Equations. The non-hydrostatic models for water waves are classified as either single-layer models (two-dimensional depth-integrated non-hydrostatic) or multi-layer models (three-dimensional non-hydrostatic), depending on the number of layers in the vertical discretization. The capabilities of the non-hydrostatic models for water waves with single or multiple layers in the vertical direction has been researched by [PERSON] and [PERSON] [77]; [PERSON] et al. [78]; [PERSON] and [PERSON] [79] and [PERSON] et al. [80], among others. The non-hydrostatic D/C Galerkin model is constructed using a combination of discontinuous and continuous Galerkin methods, where the depth-integrated non-hydrostatic Equations are separated into hydrostatic and non-hydrostatic parts. The hydrostatic part corresponds to the depth-integrated shallow water  
Equations and is solved with a discontinuous Galerkin FEM for the simulation of discontinuous flows, wave breaking and run-up. The non-hydrostatic part results in a Poisson-type equation where the non-hydrostatic pressure is resolved by a continuous Galerkin finite element method for the modeling of wave propagation and transformation. The model utilizes linear quadrilateral finite elements for horizontal velocities, water surface elevations and non-hydrostatic pressures that permit local refinement and complex boundaries. The governing Equations in the conservation form for depth-integrated, non-hydrostatic flow in the Cartesian coordinates system (\(x\), \(y\) and \(z\)) are:  
\[\frac{\partial(HU)}{\partial t}+\frac{\partial(HUI)}{\partial x}+\frac{ \partial(HUV)}{\partial y}=-gH\frac{\partial\xi}{\partial x}-\frac{g^{n2}U \sqrt{U^{2}+V^{2}}}{H^{\frac{n}{3}}}-\frac{H}{2\rho}\frac{\partial q_{b}}{ \partial x}-\frac{q_{b}}{2\rho}\frac{\partial(\xi-h)}{\partial x} \tag{8}\]  
\[\frac{\partial(HV)}{\partial t}+\frac{\partial(HUV)}{\partial x}+\frac{ \partial(HVV)}{\partial y}=-gH\frac{\partial\xi}{\partial y}-\frac{g^{n2}V \sqrt{U^{2}+V^{2}}}{H^{\frac{n}{3}}}-\frac{H}{2\rho}\frac{\partial q_{b}}{ \partial y}-\frac{q_{b}}{2\rho}\frac{\partial(\xi-h)}{\partial y} \tag{9}\]  
\[\frac{\partial H}{\partial t}+\frac{\partial(UH)}{\partial x}+\frac{\partial( VH)}{\partial y}=0 \tag{10}\]  
\[\frac{\partial W}{\partial t}=\frac{q_{b}}{\rho H} \tag{11}\]  
where \(U\), \(V\) and \(W\) are the depth-averaged velocity components in the \(x\), \(y\) and z directions; \(\rho\) is the water density; \(n\) is Manning's roughness coefficient; \(g\) is the gravitational acceleration. The flow depth is \(H=\xi+h\), where \(\xi\) is the surface elevation measured from the still water level and \(h\) is the water depth measured from the still water level. A linear distribution is assumed in the vertical direction for both the non-hydrostatic pressures and for the vertical velocities. The non-hydrostatic pressure on the free surface is assumed as zero and, in the bottom, as \(q_{b}\).