Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit

Abstract

Abstract. We prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1-Wasserstein topology (respectively in L 1 loc ) to the unique Kruzkov entropy solution of the conservation law. The initial data are taken in L ∞ , nonnegative, and with compact support, hence we are able to handle densities with vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications, e.g. in the Lighthill-Whitham-Richards model for traffic flow) with possible degenerate slope near the vacuum state. The proof of the result is based on discrete BV estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the regularizing effect L ∞ → BV for nonlinear scalar conservation laws is intrinsic of the discrete model. Keywords: Micro-macro limit and Scalar conservation laws and Follow-the-leader models and Oleinik condition and Entropy solutions and Particle method. 35L65 and 35L45 and 90B20 and 65N75 and 82C22 . AMS Subject classification:

The Aw–Rascle Traffic Model: Enskog-Type Kinetic Derivation and Generalisations

“…We consider now a space non-homogeneous scenario and we report results for the Boltzmann-type description of traffic flow coming from (10) under the binary interaction model (7) along with the quasi-invariant scaling (13). We start by giving the details of the discretisation technique.…”

“…Figure 5: Test 3. Left: density and mean speed at the computational time t = 6 obtained with the hydrodynamic model (20) and the Boltzmann-type kinetic model (10) with D = 0 in the binary interactions (7). Right: kinetic distribution in the phase space.…”

“…More recently, also the direct link between follow-the-leader microscopic models and the Aw-Rascle macroscopic model has been explored. In particular, in [9,10] the authors prove that the trajectories of the former converge, in the 1-Wasserstein metric, to the unique entropy solution of the latter when a suitable large particle limit is considered. Their strategy consists in interpreting the follow-the-leader model as a discrete Lagrangian approximation of the target macroscopic model.…”

Traveling waves for nonlocal models of traffic flow

“…where f (ρ) = ρv(ρ), (1.12) has been studied in various papers [18,25,26]. For the nonlocal model (1.1), existence of entropy weak solutions for the Cauchy problem was proved in [9] by utilizing the convergence of a finite difference scheme, and in [24] by means of a finite volume scheme.…”

An interface-free multi-scale multi-order model for traffic flow

“…Aw et al [1], Greenberg [28], and Di Francesco et al [17] investigated the many-particle limit in the framework of second-order traffic models, deriving the macroscopic Aw-Rascle-Zhang (ARZ) model [2,51] from a particular second-order microscopic follow-the-leader (FtL) model [30,43]. Instead Colombo and Rossi [9], Rossi [47], Di Francesco and Rosini [18], and Di Francesco et al [16] investigated the many-particle limit in the framework of first-order traffic models, deriving the macroscopic Lighthill-Whitham-Richards (LWR) model [39,45] as the limit of a first-order FtL model. Let us also mention the papers by Forcadel et al [24], Forcadel and Salazar [23] which investigate the many-particle limit exploiting the link between conservation laws and Hamilton-Jacobi equations.…”