Stability of stationary solutions of singular systems of balance laws

TL;DR

This paper develops a relative entropy framework to analyze the stability of stationary solutions in singular hyperbolic PDE systems, including those with discontinuous flux and non-conservative products, applicable to fluid mechanics models like shallow-water equations.

Contribution

It introduces a novel relative entropy method for stability analysis of singular systems with non-conservative terms, applicable to multidimensional unstructured mesh schemes.

Findings

Proves stability of certain stationary states within entropy weak solutions.

Establishes asymptotic stability of discrete stationary solutions in numerical schemes.

Applies to non-strictly hyperbolic and multidimensional systems.

Abstract

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.