Article
Mathematics, Applied
Alessandra Jannelli, Natale Manganaro, Alessandra Rizzo
Summary: This paper examines the well-known second order Aw-Rascle nonhomogeneous system that describes traffic flows. Using the method of differential constraints, a suitable reduction procedure is developed to solve a class of Riemann problems in traffic flow theory. The general solution of the Riemann problem, including shock waves, contact discontinuities, and generalized rarefaction waves, is obtained for a given source term. The interaction between a shock wave and a generalized rarefaction wave is also studied, and a related generalized Riemann problem is solved. Numerical results agree with the exact analytical solution.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Tingting Chen, Weifeng Jiang, Tong Li
Summary: This paper investigates the stability of the Riemann problem for the improved Aw-Rascle-Zhang model, which describes the formation and dynamics of traffic jams. By constructing classical Riemann solutions and analyzing wave interactions, the existence and uniqueness of delta-shock wave in this model are proven, leading to stability analysis of the Riemann problem.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mechanics
Weifeng Jiang, Tingting Chen, Tong Li, Zhen Wang
Summary: We study the cavitation and concentration of Riemann solutions for the improved Aw-Rascle-Zhang model in dusty gas with a non-genuinely nonlinear field. We construct the Riemann solutions containing composite waves using Liu-entropy condition, investigate the limits of inflection points and tangent points along the 1-family wave curve, and observe the formation of d-shock wave and vacuum as pressure vanishes. We conclude that the limit of Riemann solutions of the IARZ model is not the Riemann solutions of the limit of the IARZ model, consistent with the work of C. Shen and M. Sun.
Article
Physics, Mathematical
Giacomo Dimarco, Andrea Tosin, Mattia Zanella
Summary: In this paper, second order hydrodynamic traffic models are derived from kinetic-controlled equations for driver-assist vehicles, taking into account two main control strategies. The presence of driver-assist vehicles leads to an aggregate homogenisation of mean flow speed, optimizing flow and traffic stabilisation.
JOURNAL OF STATISTICAL PHYSICS
(2022)
Article
Mathematics
Weifeng Jiang, Tingting Chen, Tong Li, Zhen Wang
Summary: In this paper, we investigate the Radon measure initial value problem for the non-isentropic improved Aw-Rascle-Zhang model. We construct the Riemann solutions, including elementary waves and delta-shock waves, for arbitrary convex F(u) in this model using the method of generalized characteristic analysis. By considering the limit of solutions for initial data with three piece-wise constants, we obtain the constructive solutions for initial data containing the Dirac measure. Additionally, we analyze various wave interactions, including the interactions of delta-shock waves with elementary waves.
ACTA MATHEMATICA SCIENTIA
(2023)
Article
Mathematics
Andronikos Paliathanasis, Peter G. L. Leach
Summary: We extend our analysis on the Lie symmetries in fluid dynamics to macroscopic traffic estimation models. Specifically, we study the Aw-Rascle-Zhang model, which consists of two hyperbolic first-order partial differential equations. We determine the Lie symmetries, the one-dimensional optimal system, and the corresponding Lie invariants. We find that the admitted Lie symmetries form the four-dimensional Lie algebra A(4,12). The resulting one-dimensional optimal system is composed of seven one-dimensional Lie algebras. We use the Lie symmetries to define similarity transformations and derive new analytic solutions for the traffic model, discussing the qualitative behavior of the solutions.
Article
Mathematics, Applied
Nilasis Chaudhuri, Piotr Gwiazda, Ewelina Zatorska
Summary: We study the multi-dimensional generalization of the Aw-Rascle system for vehicular traffic. We prove the existence of global-in-time measure-valued solutions for arbitrary large initial data and periodic boundary conditions. Using the relative energy technique, we also show that the measure-valued solutions coincide with the classical solutions as long as they exist.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Weifeng Jiang, Tingting Chen, Tong Li, Zhen Wang
Summary: This study focuses on the wave interactions of an improved Aw-Rascle-Zhang model with a non-genuinely nonlinear field. By analyzing the Riemann solutions, the wave interactions between single elementary waves involving the composite wave are examined. The presence of a non-genuinely nonlinear field leads to the discovery of new phenomena, such as the ability of rarefaction waves to penetrate shock waves and the transformation of compression waves to rarefaction waves during the interaction with a contact discontinuity. Furthermore, the weak solutions of this model with three piecewise constant states are constructed based on the results of single wave interactions, providing insights for eliminating phantom traffic jams.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2023)
Article
Engineering, Civil
Rongyong Zhao, Dong Wang, Yan Wang, Chuanfeng Han, Ping Jia, Cuiling Li, Yunlong Ma
Summary: This study modified the Aw-Rascle traffic flow model to investigate crowd evacuation at T-shaped street junctions, providing suggestions for preventing stampedes and validating the model through numerical simulations.
IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
(2021)
Article
Mechanics
Shuai Fan, Yu Zhang
Summary: The Riemann problem and wave interactions are investigated for an inhomogeneous Aw-Rascle (AR) traffic flow model with extended Chaplygin gas pressure. The Riemann problem with two piecewise constants as initial data is solved and two different types of Riemann solutions involving rarefaction wave, shock wave, and contact discontinuity are obtained. The interactions of waves are analyzed by studying the Riemann problem with three-piecewise-constant initial data, and the global structures of Riemann solutions are established. The stability of solution under small perturbation of initial data is also briefly discussed.
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS
(2023)
Article
Mathematical & Computational Biology
Stephan Gerster, Michael Herty, Elisa Iacomini
Summary: This study investigates the propagation of uncertainties in the Aw-Rascle-Zhang model using wavelet-based series expansions and stochastic Galerkin formulations. Stabilization results are obtained when the system is relaxed to a first-order model, as illustrated by computational tests.
MATHEMATICAL BIOSCIENCES AND ENGINEERING
(2021)
Article
Mathematics, Applied
Qinglong Zhang, Wancheng Sheng
Summary: This paper explores the interaction of elementary waves in the AR traffic flow model with variable lane width, discussing the interactions of stationary wave with rarefaction wave, shock wave, and contact discontinuity. Solutions are constructed globally in the phase plane, potentially aiding in addressing traffic jam issues in future studies.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2021)
Article
Mathematics
Irena Strnad, Rok Marsetic
Summary: This paper presents a numerical variable speed limit (VSL) control method for a motorway, using a non-equilibrium continuum traffic model. The method includes macroscopic simulation, cost function introduction, and numerical optimization with a differential evolution algorithm. By employing a numerical solution scheme, it overcomes the limitations of basic continuum models and improves the accuracy of traffic modeling in control strategies.
Article
Multidisciplinary Sciences
Andronikos Paliathanasis
Summary: A detailed symmetry analysis is conducted for a microscopic model of traffic flow in two-lane motorways. The model is an extension of the Aw-Rascle theory and describes flow parameters using first-order partial differential equations. The model is expressed in terms of Euler and Lagrange variables, and different Lie algebras and optimal systems are found for each variable set. The Lie symmetries are then used to derive new closed-form solutions.
Article
Mathematics, Applied
Shouqiong Sheng, Zhiqiang Shao
Summary: This paper studies the concentration phenomenon and the formation of delta shock wave in the vanishing adiabatic exponent limit of Riemann solutions to the Aw-Rascle traffic model. It is proven that as the adiabatic exponent tends to zero, the limit of solutions tends to a special delta-shock rather than the classical one in zero pressure gas dynamics. By considering a perturbed Aw-Rascle model, it is rigorously proved that as the adiabatic exponent tends to one, any Riemann solution containing two shock waves tends to a delta-shock to the zero pressure gas dynamics. Representative numerical simulations are also presented to confirm the theoretical analysis.
ASYMPTOTIC ANALYSIS
(2022)
Article
Economics
Saeed Mohammadian, Zuduo Zheng, Md Mazharul Haque, Ashish Bhaskar
Summary: This paper conducts a comprehensive benchmarking study on single-pipe continuum models for freeway traffic, selecting the best representative models for assessment based on traffic data from the German A5 autobahn.
TRANSPORTATION RESEARCH PART B-METHODOLOGICAL
(2021)
Article
Computer Science, Interdisciplinary Applications
Saeed Mohammadian, Abolfazl Mohammadzadeh Moghaddam, Ali Sahaf
Summary: This paper investigates the performances of three approximate Riemann solvers in traffic flow models, finding that Rusanov and HLL solvers are more suitable for certain models. Additionally, the paper analyzes the effects of combining different spatial schemes, showing variations in handling numerical diffusions and oscillations.
COMPUTERS & FLUIDS
(2021)
Article
Public, Environmental & Occupational Health
Saeed Mohammadian, Md. Mazharul Haque, Zuduo Zheng, Ashish Bhaskar
Summary: A flexible conflict-based framework is proposed in this study to extract safety information from freeway macroscopic traffic state variables by utilizing the information from all underlying car-following interactions. The proportion of stopping distance is shown to be more desirable than several event-based conflict measures for describing time spent in conflict based on macroscopic state variables. This hybrid methodological framework combines probabilistic and machine learning models to develop relationships between safety and macroscopic state variables within a flexible conflict-based safety assessment framework.
ANALYTIC METHODS IN ACCIDENT RESEARCH
(2021)
