  The stability of stationary solutions of first-order systems of PDE's are
considered. They may include some singular geometric terms, leading to
discontinuous flux and non-conservative products. Based on several examples in
Fluid Mechanics, we assume that these systems are endowed with a partially
convex entropy. We first construct an associated relative entropy which allows
to compare two states which share the same geometric data. This way, we are
able to prove the stability of some stationary states within entropy weak
solutions. This result applies for instance to the shallow-water equations with
bathymetry. Besides, this relative entropy can be used to study finite volume
schemes which are entropy-stable and well-balanced, and due to the numerical
dissipation inherent to these methods, asymptotic stability of discrete
stationary solutions is obtained. This analysis does not make us of any
specific definition of the non-conservative products, applies to non-strictly
hyperbolic systems, and is fully multidimensional with unstructured meshes for
the numerical methods.

Analysis of PDEs;Numerical Analysis