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

TL;DR

This paper derives first order convection-diffusion traffic flow models from microscopic follow-the-leader models with reaction time, capturing both vehicular and pedestrian dynamics and matching real traffic data.

Contribution

It introduces a novel macroscopic model derived from microscopic delayed follow-the-leader models, incorporating reaction time effects in traffic flow modeling.

Findings

The macroscopic model accurately captures microscopic dynamics.

Reaction time influences the transition to stop-and-go behavior.

The model's scattering width aligns with real traffic data.

Abstract

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.