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In this study, we analyse the famous Aw-Rascle system in which the difference between the actual and the desired velocities (the offset function) is a gradient of a singular function of the density. This leads to a dissipation in the…

Analysis of PDEs · Mathematics 2022-09-27 N Chaudhuri , L Navoret , Charlotte Perrin , E Zatorska

The ability of existing headway distributions to accurately reflect the diverse behaviors and characteristics in heterogeneous traffic (different types of vehicles) and mixed traffic (human-driven vehicles with autonomous vehicles) is…

Applications · Statistics 2025-11-06 Natchaphon Leungbootnak , Zihao Li , Zihang Wei , Dominique Lord , Yunlong Zhang

Traditionally, the Lighthill-Whitham-Richards (LWR) models for homogeneous and inhomogeneous roads have been analyzed in flux-density space with the fundamental diagram of the flux-density relation. In this paper, we present a new framework…

Analysis of PDEs · Mathematics 2010-05-26 Wen-Long Jin , Liang Chen , Elbridge Gerry Puckett

Building on recent advances in scientific machine learning and generative modeling for computational fluid dynamics, we propose a conditional score-based diffusion model designed for multi-scenarios fluid flow prediction. Our model…

Machine Learning · Computer Science 2025-06-02 Wilfried Genuist , Éric Savin , Filippo Gatti , Didier Clouteau

In this paper we prove the local-in-time existence of regular solutions to dissipative Aw-Rascle system with the offset equal to gradient of some increasing and regular function of density. It is a mixed degenerate parabolic-hyperbolic…

Analysis of PDEs · Mathematics 2024-11-06 Nilasis Chaudhuri , Tomasz Piasecki , Ewelina Zatorska

In this paper, we investigate a nonlocal traffic flow model based on a scalar conservation law, where a stochastic velocity function is assumed. In addition to the modeling, theoretical properties of the stochastic nonlocal model are…

Numerical Analysis · Mathematics 2024-07-04 Timo Böhme , Simone Göttlich , Andreas Neuenkirch

In this paper, we consider the variable-order time fractional mobile/immobile diffusion (TF-MID) equation in two-dimensional spatial domain, where the fractional order $\alpha(t)$ satisfies $0<\alpha_{*}\leq \alpha(t)\leq \alpha^{*}<1$. We…

Numerical Analysis · Mathematics 2023-10-05 Xiao Ye , Jun Liu , Bingyin Zhang , Hongfei Fu , Yue Liu

We describe a fourth-order accurate finite-difference time-domain scheme for solving dispersive Maxwell's equations with nonlinear multi-level carrier kinetics models. The scheme is based on an efficient single-step three time-level…

The classical approach to linking lattice dynamics properties to continuum equations of motion, the "method of long waves," is extended to include higher order terms. The additional terms account for non-local and non-linear effects. In the…

Computational Physics · Physics 2018-09-05 Zhijie Xu , R. C. Picu , J. Fish

We propose a new model for multi-lane traffic with moving bottlenecks, e.g., autonomous vehicles (AV). It consists of a system of balance laws for traffic in each lane, coupled in the source terms for lane changing, and fully coupled to…

Analysis of PDEs · Mathematics 2024-04-29 Paola Goatin , Benedetto Piccoli

The basic conceptual picture and theoretical basis for development of transport equations in porous media are examined. The general form of the governing equations is derived for conservative chemical transport in heterogeneous geological…

Statistical Mechanics · Physics 2015-06-24 Brian Berkowitz , Joseph Klafter , Ralf Metzler , Harvey Scher

The paper provides results for a non-standard, hyperbolic, 1-D, nonlinear traffic flow model on a bounded domain. The model consists of two first-order PDEs with a dynamic boundary condition that involves the time derivative of the…

Optimization and Control · Mathematics 2017-07-10 Iasson Karafyllis , Nikolaos Bekiaris-Liberis , Markos Papageorgiou

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

We study a generalized Follow-the-Leader model where the driver considers the position of an arbitrary but finite number of vehicles ahead, as well as the position of the vehicle directly behind the driver. It is proved that this model…

Analysis of PDEs · Mathematics 2024-05-08 Helge Holden , Nils Henrik Risebro

The aim of this paper is to discuss the appropriate modelling of in- and outflow boundary conditions for nonlinear drift-diffusion models for the transport of particles including size exclusion and their effect on the behaviour of…

Analysis of PDEs · Mathematics 2016-11-03 Martin Burger , Jan-Frederik Pietschmann

The traffic modelling often keeps the mesoscopic scale in the theoretical sphere because the integro-differential nature of its equations. In the present work we suggest to use the lattice Boltzmann method to overcome these difficulties. In…

Physics and Society · Physics 2018-10-01 Romain Noël , Laurent Navarro , Guy Courbebaisse

In this paper, we develop data-driven closure/correction terms to increase the pressure and velocity accuracy of reduced order models (ROMs) for fluid flows. Specifically, we propose the first pressure-based data-driven variational…

Numerical Analysis · Mathematics 2023-01-25 Anna Ivagnes , Giovanni Stabile , Andrea Mola , Traian Iliescu , Gianluigi Rozza

We propose a Physics Informed Learning framework for reconstructing traffic density from sparse trajectory data. The approach combines a second-order Aw-Rascle and Zhang model with a first-order training stage to estimate the equilibrium…

Systems and Control · Electrical Eng. & Systems 2026-04-10 S. Betancur Giraldo , J. Mårtensson , M. Barreau

Traffic on a circular road is described by dynamic programming equations associated to optimal control problems. By solving the equations analytically, we derive the relation between the average car density and the average car flow, known…

Optimization and Control · Mathematics 2010-02-11 Nadir Farhi

This work presents a comprehensive framework for enhanced diffusion modeling in fluid-structure interactions by combining the Immersed Boundary Method (IBM) with stochastic trajectories and high-order spectral boundary conditions. Using…

Analysis of PDEs · Mathematics 2024-10-31 Rômulo Damasclin Chaves dos Santos , Jorge Henrique de Oliveira Sales