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

Abstract This article proposes a second-order MUSCL-type central scheme of the Nessyahu–Tadmor kind for a class of scalar conservation laws with discontinuous flux and presents its convergence analysis. Since solutions to problems with discontinuous flux generally do not belong to the space of bounded variation, 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. To the best of our knowledge, this work provides the first rigorous convergence analysis for a MUSCL-type second-order scheme for conservation laws with discontinuous flux.