Haar method, averaged matrix, wave cancellations, and L1 stability for hyperbolic systems

TL;DR

This paper introduces a novel Haar-Holmgren method for controlling the L1 distance between entropy solutions of nonlinear hyperbolic systems, including those with discontinuous coefficients, ensuring stability and uniqueness.

Contribution

It extends existing methods to handle discontinuous coefficients and general flux systems, providing new stability results for solutions with small total variation.

Findings

Proves L1 stability of solutions generated by Glimm or front tracking schemes.

Shows averaged matrices have no rarefaction-shocks, aiding in stability analysis.

Extends stability results to fluid dynamics equations and non-genuinely nonlinear systems.

Abstract

We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L1 distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks -- which are recognized as a source for non-uniqueness and instability. Our Haar-Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It extends the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L1 norm upon their initial data, by exhibiting an L1 functional controling the distance between two solutions.