# Equations and discretization

This example shows a simplified implementation of the **Shallow-Water** equations through a simple Euler temporal scheme and spatial Rusanov scheme. This example does not consider any friction or Coriolis force. To get a complete model of the **Shallow-Water** equations, the **Tolosa-sw** model and its implementation are further described in the **Tolosa-sw** tab.

## Equations¶

The Non-linear Shallow Water equations (NSW) are formulated below with conservative variables:

\(h = h(\mathbf{x}, t)\) stands for the water depth and \(\mathbf{u} = \mathbf{u}(\mathbf{x}, t)\) the horizontal velocity, functions of the space and time variables. The pressure force is taken into account through the quantity \(\Phi = g(h+z)\), \(g\) being the gravity constant and \(z = z(\mathbf{x})\) standing for a parametrization of the topography.

Note

Here, the friction and Coriolis force are not considered.

## Discretization¶

This example will present a numerical discretization of these equations through the use of **Tolosa-lib**. The discretization is written with a temporal **Euler** scheme and a spatial **Rusanov** scheme.