An overview of non-local traffic flow models

We give an overview of mathematical traffic flow models with non-local velocity. More precisely, we consider conservation laws with flux functions depending on an integral evaluation of the density of vehicles through a convolution product. We summarize the analytical results recently obtained for this kind of models and we provide some numerical simulations illustrating the behavior of different groups of drivers or vehicles

Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity

International audienceWe consider an extension of the traffic flow model proposed by Lighthill, Whitham and Richards, in which the mean velocity depends on a weighted mean of the downstream traffic density. We prove well-posedness and a regularity result for entropy weak solutions of the corresponding Cauchy problem, and use a finite volume central scheme to compute approximate solutions. We perform numerical tests to illustrate the theoretical results and to investigate the limit as the convolution kernel tends to a Dirac delta function. 1. Introduction. Macroscopic traffic flow models provide nowadays a validated and powerful approach to simulate and manage traffic on road networks (we refer the interested reader to [25] for an overview of modelling approaches and practical applications and to [11] for a detailed review of the mathematical theory of macroscopic models on networks). These models are based on hyperbolic equations derived from fluid dynamics, and describe the spatio-temporal evolution of macroscopic quantities like vehicle density and mean velocity. One of the seminal macroscopic models was introduced in the mid 1950s by Lighthill and Whitham [21] and Richards [23], who proposed to complete the one-dimensional mass conservation equation ∂ t ρ(t, x) + ∂ x f (t, x) = 0 with a closure relation between speed and density, leading to the fundamental diagram f (t, x) = f (ρ(t, x)) = ρ(t, x)v(ρ(t, x)), which can be derived from empirical speed-density or flow-density data. The classical LWR model therefore reads ∂ t ρ(t, x) + ∂ x (ρ(t, x)v(ρ(t, x))) = 0, x ∈ R, t > 0

Accelerated Kinetic Monte Carlo methods for general nonlocal traffic flow models

This paper presents a class of one-dimensional cellular automata (CA) models on traffic flows, featuring nonlocal look-ahead interactions. We develop kinetic Monte Carlo (KMC) algorithms to simulate the dynamics. The standard KMC method can be inefficient for models with global interactions. We design an accelerated KMC method to reduce the computational complexity in the evaluation of the nonlocal transition rates. We investigate several numerical experiments to demonstrate the efficiency of the accelerated algorithm, and obtain the fundamental diagrams of the dynamics under various parameter settings.Comment: 22 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:2001.1001

High-resolution central-upwind scheme for second-order macroscopic traffic flow models

Traffic flow models are important tools for traffic management applications such as traffic incident detection and traffic control. In this paper, we propose a novel numerical approximation method for second-order macroscopic traffic flow models. The method is based on the semi-discrete central-upwind numerical flux and high-order reconstructions for spatial discretizations. We then apply the designed high-resolution schemes to three representative types of second-order traffic flow models and perform a variety of numerical experiments to validate the proposed methods. The simulation results illustrate the effectiveness, simplicity and universality of the central-upwind scheme as numerical approximation method for macroscopic traffic flow models