Convergence of a second-order central scheme for conservation laws with discontinuous flux

In this article, we propose a Nessyahu-Tadmor-type second-order central scheme for a class of scalar conservation laws with discontinuous flux and present its convergence analysis. Since solutions to problems with discontinuous flux typically do not belong to the space of bounded variation (BV), we employ the theory of compensated compactness as the main tool for the convergence of approximate solutions. A central component of the analysis involves establishing the maximum principle and the $\mathrm{W}^{-1,2}_{\mathrm{loc}}$ compactness of the approximate solutions, the latter achieved through the derivation of several essential estimates. Finally, by introducing a mesh-dependent correction term in the slope limiter, we show that the numerical solutions generated by the the proposed second-order scheme converge to the entropy solution.