# American Institute of Mathematical Sciences

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January  2020, 40(1): 233-266. doi: 10.3934/dcds.2020010

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

 Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Via Vetoio 1, Coppito, I-67100 L'Aquila, Italy

Received  January 2019 Revised  May 2019 Published  October 2019

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.

Citation: Marco Di Francesco, Graziano Stivaletta. Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 233-266. doi: 10.3934/dcds.2020010
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