Equations and Modelisation

Equations

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

where stands for the water depth, for the horizontal velocity and for the bottom elevation. is the Coriolis force and the friction source term.

For hydraulic simulations, the friction source term is defined by the Manning-Strickler friction source term:

with depending on the uniform Manning’s roughness coefficient.

For oceanic simulations, the friction source term can be written as:

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A time-splitting scheme is used to treat the friction source term, then we focus here on the resolution of:

The semi-implicit time step resolution writes:

which easily gives the final expression of the velocity:

The full implicit time step resolution writes:

Remarking that , the solution is given by a quadratic equation:

with the final solution:

The semi-implicit time step resolution writes:

which easily gives the final expression of the velocity:

The full implicit time step resolution writes:

Again, remarking that , the solution is given by a quadratic equation:

with the final solution:

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