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The formation and development of oscillations is an important traffic flow phenomenon. Recent studies found that along a vehicle platoon described in the Lagrangian specification, traffic oscillations grow in a concave way. Since stationary…

Pattern Formation and Solitons · Physics 2018-10-22 Junfang Tian , Martin Treiber , Rui Jiang , Xiaopeng Li , Bin Jia , Ziyou Gao

Most car-following models show a transition from laminar to ``congested'' flow and vice versa. Deterministic models often have a density range where a disturbance needs a sufficiently large critical amplitude to move the flow from the…

Statistical Mechanics · Physics 2007-05-23 Kai Nagel , Christopher Kayatz , Peter Wagner

The Eulerian-Lagrangian approach based on Large-Eddy Simulation (LES) is one of the most promising and viable numerical tools to study turbulent dispersed flows when the computational cost of Direct Numerical Simulation (DNS) becomes too…

Fluid Dynamics · Physics 2017-06-02 Alessio Innocenti , Cristian Marchioli , Sergio Chibbaro

This contribution compares several different approaches allowing one to derive macroscopic traffic equation directly from microscopic car-following models. While it is shown that some conventional approaches lead to theoretical problems, it…

Physics and Society · Physics 2015-05-13 Dirk Helbing

The fundamental relationship of traffic flow is empirically estimated by fitting a regression curve to a cloud of observations of traffic variables. Such estimates, however, may suffer from the confounding/endogeneity bias due to omitted…

Econometrics · Economics 2021-04-07 Anupriya , Daniel J. Graham , Daniel Hörcher , Prateek Bansal

In this paper, we introduce and study one-dimensional models for the behavior of pedestrians in a narrow street or corridor. We begin at the microscopic level by formulating a stochastic cellular automata model with explicit rules for…

Probability · Mathematics 2012-09-27 Alina Chertock , Alexander Kurganov , Anthony Polizzi , Ilya Timofeyev

We propose an efficient method for reconstructing traffic density with low penetration rate of probe vehicles. Specifically, we rely on measuring only the initial and final positions of a small number of cars which are generated using…

Dynamical Systems · Mathematics 2026-02-13 Nail Baloul , Amaury Hayat , Thibault Liard , Pierre Lissy

We derive macroscopic traffic equations from specific gas-kinetic equations, dropping some of the assumptions and approximations made in previous papers. The resulting partial differential equations for the vehicle density and average…

Statistical Mechanics · Physics 2009-10-31 Martin Treiber , Ansgar Hennecke , Dirk Helbing

This contribution investigates an original stochastic approach for the emergence of stop-and-go waves in traffic flow, a collective phenomenon with significant safety and environmental implications. Using a stable nonlinear car-following…

Physics and Society · Physics 2025-12-04 Raphael Korbmacher , Parthib Khound , Antoine Tordeux , Frank Gronwald

Connections between microscopic follow-the-leader and macroscopic fluid-dynamics traffic flow models are already well understood in the case of vehicles moving on a single road. Analogous connections in the case of road networks are instead…

Dynamical Systems · Mathematics 2016-10-04 Emiliano Cristiani , Smita Sahu

Flow matching trains a neural velocity field by regression against a target velocity associated with a prescribed probability path connecting a simple initial distribution to the data distribution. A central design choice is the path…

Machine Learning · Computer Science 2026-05-21 Shukai Du , Junzhe Zhang , Yiming Li

The log-homotopy particle flow filter resolves the Bayesian update by transporting particles along a continuous trajectory in pseudo-time. However, the governing partial differential equation for the flow velocity is fundamentally…

Systems and Control · Electrical Eng. & Systems 2026-05-18 Olivér Törő , Domonkos Csuzdi , Tamás Bécsi

We investigate the existence of weak type solutions for a class of aggregation-diffusion PDEs with nonlinear mobility obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as…

Analysis of PDEs · Mathematics 2018-03-30 Simone Fagioli , Emanuela Radici

We develop a control design for stabilization of traffic flow in congested regime, based on an Aw-Rascle-Zhang-type (ARZ-type) Partial Differential Equation (PDE) model, for traffic consisting of both ACC-equipped (Adaptive Cruise…

Optimization and Control · Mathematics 2018-12-20 Nikolaos Bekiaris-Liberis , Argiris I. Delis

A fully discrete Lagrangian scheme for numerical solution of the nonlinear fourth order DLSS equation in one space dimension is analyzed. The discretization is based on the equation's gradient flow structure in the $L^2$-Wasserstein metric.…

Numerical Analysis · Mathematics 2014-10-08 Daniel Matthes , Horst Osberger

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation's gradient flow structure with respect to the Wasserstein distance. The scheme inherits…

Numerical Analysis · Mathematics 2019-02-20 Daniel Matthes , Horst Osberger

In this paper, we present a class of systems of non-local conservation laws in one space-dimension incorporating time delay, which can be used to investigate the interaction between autonomous and human-driven vehicles, each characterized…

Analysis of PDEs · Mathematics 2025-01-17 Ilaria Ciaramaglia , Paola Goatin , Gabriella Puppo

We find a further class of exact solutions to the Lighthill Whitham Richards Payne (LWRP) traffic flow equations. As before, using two consecutive Lagrangian transformations, a linearization is achieved. Next, depending on the initial…

Fluid Dynamics · Physics 2014-10-09 E. Infeld , G. Rowlands , A. A. Skorupski

We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attraction/repulsion, and we present a numerical scheme for its solution through a Lagrangian discretization. The density…

Numerical Analysis · Mathematics 2020-05-01 Hugo Leclerc , Quentin Mérigot , Filippo Santambrogio , Federico Stra

From the macroscopic viewpoint for describing the acceleration behavior of drivers, this letter presents a weighted probabilistic cellular automaton model (the WP model, for short) by introducing a kind of random acceleration probabilistic…

Physics and Society · Physics 2009-11-13 X. L. Li , H. Kuang , T. Song , S. Q. Dai , Z. P. Li
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