Top-level heading

A deterministic particle approximation for nonlinear conservation laws

Data e ora inizio evento
Data e ora fine evento
Sede

Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma

Aula
Sala di Consiglio
Speaker

Marco Di Francesco (Università de L'Aquila)

We present recent results on the deterministic particle approximation of non-linear conservation laws. In [1], the unique entropy solution to a scalar conservation law with a given initial datum in L∞ and with strictly monotone v is rigorously approximated by the empirical measure of a follow-the-leader particle system. Said result is based on a discrete version of the classical Oleinik one-sided jump condition for L∞ initial data and on a BV contraction estimate for BV initial data. The former requires some additional conditions on v, which reduces to strict concavity of the flux in case v is a power law. The convergence result also holds for the discrete density constructed from the particle system. The results in [1] have been recently extended to the Aw-Rascle-Zhang model for traffic flow in [2], where a similar BV contraction estimate has been proven, based on the interpretation of the system as a multi-population model. Finally, we shall present an extension of this technique to the Hughes model for pedestrians on a bounded interval with Dirichlet boundary conditions. In [3] we prove the rigorous convergence of a suitable adaptation of the above particle scheme to the unique entropy solution to the IBV problem for the Hughes model. Joint work with: Simone Fagioli (University of L'Aquila), Massimiliano D. Rosini (Lublin University of Technology), Giovanni Russo (University of Catania). [1] M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for rational mechanics and analysis, 217 (3) (2015), pp. 831-871. [2] M. Di Francesco, S. Fagioli, and M. D. Rosini, Many particle approximation for the Aw-Rascle-Zhang second order model for vehicular traffic, Submitted preprint. [3] M. Di Francesco, S. Fagioli, M. D. Rosini, and G. Russo, Deterministic particle approximation of the Hughes model in one space dimension, in preparation.