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Some recents results about well-balanced schemes

In a first step, we consider the delicate problem of the derivation of positive, entropy preserving and well-balanced scheme for the
shallow-water model. Several schemes were recently proposed in the literature but, in general, the well-balanced property just concerns the
steady states at rest. Here, we derive a fully well-balanced method able to exactly restore all the steady states (at rest or moving). In addition,
the scheme is proved to be positive preserving and entropy stable.
Next, we extend the well-balanced schemes to approximate the weak solutions of more sophisticated models. In this talk, we will consider the
Ripa model and the Euler equations with gravity potential. Now, the main difficulty stays in the formulation of the steady states of interest since
they are governed by nonlinear PDE and explicit formulation cannot be reached. A suitable numerical scheme is thus derived, which is able to
correctly approximate the steady states.
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