The shallow water description through the Saint-Venant system is usual for many applications (rivers flow, tidal waves, but also narrow tubes). This is an hyperbolic system, relatively simple, but that contains a source terms describing the bottom topography. Classical finite volumes schemes give a very low accuracy on such a system, especially because they do not preserv the steady states. This question has been considered by many authors who modify the most classical solvers.
In this talk we address two questions on the Saint-Venant system. First, we show how a derivation from Navier-Stokes (and not Euler) equations allow to justify the friction term, the viscosity terms and the Bousinesq coefficients. Second, we show how the kinetic approach allows a simple understanding of stiff topography and the derivation a solver for finite volumes methods which stability condition is independent of the source term. These concepts will be illustrated on realistic test cases.
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