Analysis of a central MUSCL-type scheme for conservation laws with discontinuous flux

TL;DR

This paper introduces a second-order central scheme for scalar conservation laws with discontinuous flux, proving convergence using compensated compactness and validating with numerical examples.

Contribution

The paper develops a novel second-order Nessyahu-Tadmor-type scheme for discontinuous flux problems and provides a rigorous convergence analysis using advanced compactness techniques.

Findings

Convergence of the scheme to the entropy solution is established.

The scheme maintains the maximum principle.

Numerical examples confirm theoretical results.

Abstract

In this article, we propose a second-order central scheme of the Nessyahu-Tadmor-type for a class of scalar conservation laws with discontinuous flux and present its convergence analysis. Since solutions to problems with discontinuous flux generally do not belong to the space of bounded variation (BV), we employ the theory of compensated compactness to establish the convergence of approximate solutions. A major component of our analysis involves deriving the maximum principle and showing the $\mathrm{W}^{-1,2}_{\mathrm{loc}}$ compactness of a sequence constructed from approximate solutions. The latter is achieved through the derivation of several essential estimates on the approximate solutions. Furthermore, by incorporating a mesh-dependent correction term in the slope limiter, we show that the numerical solutions generated by the proposed second-order scheme converge to the entropy solution. Finally, we validate our theoretical results by presenting numerical examples.