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Tolosa Documentation

Equations and Discretization

This example shows a simplified implementation of the Shallow-Water equations using a spatial Rusanov scheme and a temporal Euler scheme, without friction or Coriolis force. For a complete model, see Tolosa-sw.

Equations

Section titled “Equations”

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

where is the water depth, the horizontal velocity, and the pressure force with the topography.

The 2D form:

The eigenvalues of the Jacobian matrix are:

Spatial Rusanov Scheme

Section titled “Spatial Rusanov Scheme”

The Rusanov scheme1 computes fluxes by projecting onto the local outward normal :

where and .

The interface wave speeds are . The Rusanov fluxes from cell to :

Recovering Cartesian fluxes:

Temporal Euler Scheme

Section titled “Temporal Euler Scheme”

The Euler time discretization:

where and is the cell area.

Footnotes

Section titled “Footnotes”

  1. A modified Rusanov scheme for shallow water equations with topography and two phase flows, Kamel Mohamed and F. Benkhaldoun ↩