volume of an incompressible fluid with uniform and constant (in time) density, with its bottom surface in contact with a flat and rigid boundary, and free surface at pressure $p_0$. A sketch illustrating the configuration is shown in Fig. \[diagram\]; $x$ and $y$ are the horizontal coordinates, $z$ is the vertical one, $h$ is the height of the fluid column (i.e., the $z$ value at the free surface), $h_0$ is the height of the fluid column at rest, $L$ is a characteristic horizontal length, gravity acts on the $-\hat{z}$ direction and its value is given by $g$. It is assumed that $L \gg h_0$ as we are interested in shallow flows.

![The shallow water geometry considered in the simulations: $x$ and $y$ are the horizontal coordinates, and $z$ is the vertical coordinate. The surface height is $h$, with $h_0$ the height of the fluid column at rest. The fluid surface is at pressure $p_0$. $L$ is a characteristic horizontal length (assumed to be much larger than $h_0$). Gravity acts on the $-\hat{z}$ direction and its acceleration has a value of $g$. []{data-label="diagram"}](Figure1.eps){width="48.00000%"}

In the inviscid case, both the Euler equation and the incompressibility condition hold in the fluid body, $$\begin{aligned}
    