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.