Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux

This paper deals with the derivation of entropy solutions to Cauchy problems for a class of scalar conservation laws with space-density depending fluxes from systems of deterministic particles of follow-the-leader type. We consider fluxes which are product of a function of the density $v(\rho)$ and a function of the space variable $\phi(x)$. We cover four distinct cases in terms of the sign of $\phi$, including cases in which the latter is not constant. The convergence result relies on a local maximum principle and on a uniform $BV$ estimate for the approximating density.