Convergence rates of monotone schemes for conservation laws with discontinuous flux

TL;DR

This paper establishes the first known convergence rate for monotone finite volume schemes applied to scalar conservation laws with discontinuous flux, demonstrating a rate of .5 in  norm under specific conditions.

Contribution

It provides the first proof of convergence rates for numerical methods solving conservation laws with discontinuous, nonlinear flux functions.

Findings

Convergence rate of .5 in  norm for the schemes.

Numerical experiments confirming theoretical results.

Extension of convergence rate proofs to initial and boundary value problems.

Abstract

We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of $\sqrt{\Delta x}$ in $\mathrm{L}^1$, whenever the flux is strictly monotone in $u$ and the spatial dependency of the flux is piecewise constant with finitely many discontinuities. We also present numerical experiments to illustrate the main result. To the best of our knowledge, this is the first proof of any type of convergence rate for numerical methods for conservation laws with discontinuous, nonlinear flux. Our proof relies on convergence rates for conservation laws with initial and boundary value data. Since those are not readily available in the literature we establish convergence rates in that case en passant in the Appendix.