Abstract
We consider the Saint-Venant system for shallow water flows with nonflat bottom. In past years, efficient well-balanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steady state reconstruction that allows one to derive a subsonic-well-balanced scheme, preserving exactly all the subsonic steady states. It generalizes the now well-known hydrostatic solver, and like the latter it preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality. An application to the Euler-Poisson system is proposed.
Highlights
We consider the classical Saint-Venant system for shallow water flows with topography
It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, oceans when completed with a Coriolis term, and granular flows when completed with friction
The Saint Venant system describes the evolution of the water height ρ(t, x) and the velocity u(t, x) in the horizontal direction, of a thin layer of water flowing over a slowly varying topography
Summary
We consider the classical Saint-Venant system for shallow water flows with topography. One has to use the so called well-balanced schemes, that properly balance the fluxes and the source at the level of each interface. To the well-balanced property, the difficulty is to haves schemes that satisfy very natural properties such as conservativity of the water height ρ, nonnegativity of ρ, the ability to compute dry states ρ = 0 and transcritical flows when the Jacobian matrix F of the flux function becomes singular, and eventually to satisfy a discrete entropy inequality. Shallow water, subsonic reconstruction, subsonic steady states, well-balanced scheme, semi-discrete entropy inequality. Note that a solver proposed in [9] is able to maintain all the steady states, but it is not satisfying a discrete entropy inequality
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