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Nicolas Seguin
Stability of stationary solutions of singular systems of balance laws
Confluentes Mathematici, 10 no. 2 (2018), p. 93-112, doi: 10.5802/cml.52
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Class. Math.: 35L60, 35B35, 35B25, 65M08
Keywords: Hyperbolic systems, stationary state, stability, relative entropy, non-conservative systems, finite volume schemes, well-balanced schemes.

Résumé - Abstract

The stability of stationary solutions of first-order systems of PDE’s is considered. The systems under investigation may include 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. Let us stress that these solutions are only required to have a bounded total variation, i.e. they can be discontinuous. 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.

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