From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models

In this work, we derive first order continuum traffic flow models from a microscopic delayed follow-the-leader model. Those are applicable in the context of vehicular traffic flow as well as pedestrian traffic flow. The microscopic model is based on an optimal velocity function and a reaction time parameter. The corresponding macroscopic formulations in Eulerian or Lagrangian coordinates result in first order convection-diffusion equations. More precisely, the convection is described by the optimal velocity while the diffusion term depends on the reaction time. A linear stability analysis for homogeneous solutions of both continuous and discrete models are provided. The conditions match the ones of the car-following model for specific values of the space discretization. The behavior of the novel model is illustrated thanks to numerical simulations. Transitions to collision-free self-sustained stop-and-go dynamics are obtained if the reaction time is sufficiently large. The results show that the dynamics of the microscopic model can be well captured by the macroscopic equations. For non--zero reaction times we observe a scattered fundamental diagram. The scattering width is compared to real pedestrian and road traffic data

Two-way multi-lane traffic model for pedestrians in corridors

We extend the Aw-Rascle macroscopic model of car traffic into a two-way multi-lane model of pedestrian traffic. Within this model, we propose a technique for the handling of the congestion constraint, i.e. the fact that the pedestrian density cannot exceed a maximal density corresponding to contact between pedestrians. In a first step, we propose a singularly perturbed pressure relation which models the fact that the pedestrian velocity is considerably reduced, if not blocked, at congestion. In a second step, we carry over the singular limit into the model and show that abrupt transitions between compressible flow (in the uncongested regions) to incompressible flow (in congested regions) occur. We also investigate the hyperbolicity of the two-way models and show that they can lose their hyperbolicity in some cases. We study a diffusive correction of these models and discuss the characteristic time and length scales of the instability

How can macroscopic models reveal self-organization in traffic flow?

In this paper we propose a new modeling technique for vehicular traffic flow, designed for capturing at a macroscopic level some effects, due to the microscopic granularity of the flow of cars, which would be lost with a purely continuous approach. The starting point is a multiscale method for pedestrian modeling, recently introduced in Cristiani et al., Multiscale Model. Simul., 2011, in which measure-theoretic tools are used to manage the microscopic and the macroscopic scales under a unique framework. In the resulting coupled model the two scales coexist and share information, in the sense that the same system is simultaneously described from both a discrete (microscopic) and a continuous (macroscopic) perspective. This way it is possible to perform numerical simulations in which the single trajectories and the average density of the moving agents affect each other. Such a method is here revisited in order to deal with multi-population traffic flow on networks. For illustrative purposes, we focus on the simple case of the intersection of two roads. By exploiting one of the main features of the multiscale method, namely its dimension-independence, we treat one-dimensional roads and two-dimensional junctions in a natural way, without referring to classical network theory. Furthermore, thanks to the coupling between the microscopic and the macroscopic scales, we model the continuous flow of cars without losing the right amount of granularity, which characterizes the real physical system and triggers self-organization effects, such as, for example, the oscillatory patterns visible at jammed uncontrolled crossroads.Comment: 7 pages, 7 figure

Multiscale modeling of granular flows with application to crowd dynamics

In this paper a new multiscale modeling technique is proposed. It relies on a recently introduced measure-theoretic approach, which allows to manage the microscopic and the macroscopic scale under a unique framework. In the resulting coupled model the two scales coexist and share information. This allows to perform numerical simulations in which the trajectories and the density of the particles affect each other. Crowd dynamics is the motivating application throughout the paper.Comment: 30 pages, 9 figure

Multiscale modeling of granular flows with application to crowd dynamics

In this paper a new multiscale modeling technique is proposed. It relies on a recently introduced measure-theoretic approach, which allows to manage the microscopic and the macroscopic scale under a unique framework. In the resulting coupled model the two scales coexist and share information. This allows to perform numerical simulations in which the trajectories and the density of the particles affect each other. Crowd dynamics is the motivating application throughout the paper.Comment: 30 pages, 9 figure

Feedback Control of Macroscopic Crowd Dynamic Models

This paper presents design of nonlinear feedback controllers for two different macroscopic models for two- dimensional pedestrian dynamics. The models presented here are based on the laws of conservation of mass and momentum. These models have been developed by extending one-dimension macroscopic vehicle traffic flow models that use two-coupled partial deferential equations (PDEs). These models modify the vehicle traffic models so that bi-directional controlled flow is possible. Both models satisfy the conservation principle and are classified as nonlinear, time-dependent, hyperbolic PDE systems. The equations of motion in both cases are described by nonlinear partial differential equations. We address the feedback control problem for both models in the framework of partial differential equations. The objective is to synthesize nonlinear distributed feedback controllers that guarantee stability of a closed loop system