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:

\[\begin{align} \partial_{t} h + \operatorname{div}(h\mathbf{u}) &= 0 \\ \partial_{t} (h\mathbf{u}) + \operatorname{div}(h\mathbf{u}\otimes\mathbf{u}) &= -h \nabla \Phi \end{align}\]

\(h = h(\mathbf{x}, t)\) 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.