We consider the Saint-Venant system for shallow water flows, with
non-flat bottom. 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, or
granular flows when completed with friction. Numerical approximate
solutions to this system may be generated using conservative finite
volume methods, which are known to properly handle shocks and
contact discontinuities. However, in general these schemes are known
to be quite inaccurate for near steady states, as the structure of
their numerical truncation errors is generally not compatible with
exact physical steady state conditions. This difficulty can be overcome
by using the so called well-balanced schemes.
We describe a general strategy, based on a local hydrostatic
reconstruction, that allows to derive a well-balanced scheme from any
given numerical flux for the homogeneous problem.
Whenever the initial solver satisfies some classical stability
properties, it yields a simple and fast well-balanced scheme that
preserves the nonnegativity of the water height and satisfies a
semi-discrete entropy inequality.
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