Flux-stability for conservation laws with discontinuous flux and convergence rates of the front tracking method

TL;DR

This paper establishes stability and convergence rates for the front tracking method applied to scalar conservation laws with discontinuous flux, providing the first such results in this area and validating them through numerical experiments.

Contribution

It proves stability of adapted entropy solutions under flux variations and derives convergence rates for the front tracking method in discontinuous flux scenarios.

Findings

Stability of solutions with respect to flux changes.

Convergence rates for the front tracking method.

Numerical validation of theoretical results.

Abstract

We prove that adapted entropy solutions of scalar conservation laws with discontinuous flux are stable with respect to changes in the flux under the assumption that the flux is strictly monotone in u and the spatial dependency is piecewise constant with finitely many discontinuities. We use this stability result to prove a convergence rate for the front tracking method -- a numerical method which is widely used in the field of conservation laws with discontinuous flux. To the best of our knowledge, both of these results are the first of their kind in the literature on conservation laws with discontinuous flux. We also present numerical experiments verifying the convergence rate results and comparing numerical solutions computed with the front tracking method to finite volume approximations.