Traffic Flow Theory for One Dimension and Formation of Shock Speed (original) (raw)
Macroscopic Analysis Of The Viscous-Diffusive Traffic Flow Model
Mathematics in Applied Sciences and Engineering
Second-order macroscopic traffic models are characterized by a continuity equation and an acceleration equation. Convection, anticipation, relaxation, diffusion, and viscosity are the predominant features of the different classes of the acceleration equation. As a unique approach, this paper presents a new macro-model that accounts for all these dynamic speed quantities. This is done to determine the collective role of these traffic quantities in macroscopic modeling. The proposed model is solved numerically to explain some phenomena of a multilane traffic flow. It also includes a linear stability analysis. Furthermore, the evolution of speed and density wave profiles are presented under the perturbation of some parameters.
Inflow Outflow Effect and Shock Wave Analysis in a Traffic Flow Simulation
American Journal of Computational Mathematics, 2016
This paper investigates the effect of inflow, outflow and shock waves in a single lane highway traffic flow problem. A constant source term has been introduced to demonstrate the inflow and outflow. The classical Lighthill Whitham and Richards (LWR) model combined with the Greenshields model is used to obtain analytical and numerical solutions. The model is treated as an IBVP and numerical solutions are presented using Lax Friedrichs scheme. Godunov method is also used to present shock wave analysis. The numerical procedures adopted in this investigation yield results which are very much consistent with real life scenario in terms of traffic density and velocity.
Numerical Simulation of Traffic Flow via Fluid Dynamics Approach
This article deals with traffic flow simulation in a single road via fluid dynamics approach. The Lighthill-Whitham-Richards (LWR) model is used to describe traffic flow in the road which is represented by density and average speed of vehicles. Numerical approximation of the LWR model formulated as scalar conservation laws is obtained by implementing finite volume method. Several test cases are presented to simulate propagation of rarefaction and shock wave which arise in traffic flow phenomena. It has been shown that numerical results in term of density confirm the exact solutions. Numerical simulations of congestion triggered by the existence of traffic light are also discussed. The simulation results show that through adjusting the red and green light period, we can control traffic flow in traffic light location.
Physica D: Nonlinear Phenomena, 2013
Based on a Boltzmann-like traffic equation for aggressive drivers we construct in this paper a secondorder continuum traffic model which is similar to the Navier-Stokes equations for viscous fluids by applying two well-known methods of gas-kinetic theory, namely: the Chapman-Enskog method and the method of moments of Grad. The viscosity coefficient appearing in our macroscopic traffic model is not introduced in an ad hoc way-as in other second-order traffic flow models-but comes into play through the derivation of a first-order constitutive relation for the traffic pressure. Numerical simulations show that our Navier-Stokeslike traffic model satisfies the anisotropy condition and produces numerical results which are consistent with our daily experiences in real traffic.
2024
In developing countries, the quality of driving infrastructure, specifically road conditions, is often suboptimal, presenting challenges and limitations for motorists. However, current traffic flow models have limitations in addressing problems caused by poor road networks. To address this issue, a new macroscopic traffic flow model has been proposed in this study that considers meanfield velocity differences on roads with suboptimal conditions. A thorough model derivation of this new macroscopic traffic flow model is presented. The study establishes crucial stability conditions, providing profound insights into traffic dynamics across diverse scenarios. Numerical simulations are presented to demonstrate the model's ability to capture shock waves, rarefaction waves, and local cluster effects. The study results offer new insights into traffic dynamics in adverse road conditions and enforce the need to enhance road infrastructure to alleviate congestion and enhance road safety.
Macroscopic Models in Traffic Flow
Qualitative Theory of Dynamical Systems, 2008
The most known macroscopic equations for traffic flow are reviewed from a unified point of view. The analysis is done taking into account their origin, historical development and the results coming from the different treatments. We consider mainly models which describe traffic flow by means of up to three evolution equations for macroscopic variables.
Multilane analysis of a viscous second-order macroscopic traffic flow model
SN Partial Differ. Equ. Appl., 2021
Vehicular flow modeling has received much attention in the past decade due to the consequential effect of the increasing number of vehicles. A notable effect is the congestion on urban and semi-urban roads. Traffic flow models are often the first point of reference in addressing these congestion problems. In that regard, a new viscous second-order macroscopic model is presented to explore some dynamics of multilane traffic. The new model accounts for viscosity and the velocity differentials across infinitely many countable lanes. It is realized that the wave properties of the proposed model are analogous to the driving setting on a Ghanaian highway. This is followed by a mathematical condition to achieving a stable traffic flow. Moreover, the viscous model is recast into its discrete form to address interdependency among unique multiple lanes. A simulation result of an eightlane infrastructure is presented to explain this conceptualization. Keywords Viscosity Á Multilane traffic Á Macroscopic model Á Speed-density profiles Mathematics Subject Classification 35L65 Á 65M06 Á 76L05 This article is part of the section ''Applications of PDEs'' edited by Hyeonbae Kang.
Transportation Research Part B: Methodological, 1994
This paper presents a simple representation of traffic on a highway with a single entrance and exit. The representation can be used to predict traffic's evolution over time and space, including transient phenomena such as the building, propagation, and dissipation of queues. The easy-to-solve difference equations used to predict traffic's evolution are shown to be the discrete analog of the differential equations arising from a special case of the hydrodynamic model of traffic flow. The proposed method automatically generates appropriate changes in density at locations where the hydrodynamic theory would call for a shockwave; i.e., a jump in density such as those typically seen at the end of every queue. The complex side calculations required by classical methods to keep track of shockwaves are thus eliminated. The paper also shows how the equations can mimic the real-life development of stop-and-go traffic within moving queues.
International Journal for Numerical Methods in Engineering - INT J NUMER METHOD ENG, 2008
This paper firstly presents the existence and uniqueness properties of the intersection time between two neighboring shocks or between a shock and a characteristic for the analytical shock-fitting algorithm that was proposed to solve the Lighthill-Whitham-Richards (LWR) traffic flow model with a linear speeddensity relationship in accordance with the monotonicity properties of density variations along a shock, which have greatly improved the robustness of the analytical shock-fitting algorithm. Then we discuss the efficient evaluation of the measure of effectiveness (MOE) of the analytical shock-fitting algorithm. We develop explicit expressions to calculate the MOE-which is the total travel time that is incurred by travelers, within the space-time region that is encompassed by the shocks and/or characteristic lines. A numerical example is used to illustrate the effectiveness and efficiency of the proposed method compared with the numerical solutions that are obtained by a fifth-order weighted essentially non-oscillatory scheme.
On acceleration of traffic flow
13th International IEEE Conference on Intelligent Transportation Systems, 2010
The paper contributes to the derivation and analysis of accelerations in freeway traffic flow models. First, a solution based on fluid dynamics and on pure mathematical manipulations is given to express accelerations. The continuoustime acceleration is then approximated by a discrete-time equivalent. By applying continues time microscopic and macroscopic traffic flow velocity definitions, spatial and material derivatives are used to describe the continuous-time and exact changes in the velocity vector field. A forward-difference Euler method is proposed to discretize the acceleration both in time and space. For applicability purposes the use of average quantities is proposed. The finite-difference approximation by space-mean speed is shown to be consistent, and its solution is convergent to the original continuous-time form. As an alternative, the acceleration obtained from a second-order macroscopic freeway model by means of physical interpretation [1] is analyzed and found to be an appropriate discrete approximation. Comparative remarks as well as future research questions conclude the paper.