Traffic Flow Theory for One Dimension and Formation of Shock Speed (original) (raw)
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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.
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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.
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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 the Mathematical Theory of Vehicular Traffic Flow I: Fluid Dynamic and Kinetic Modelling
Mathematical Models and Methods in Applied Sciences, 2002
This review reports the existing literature on traffic flow modelling in the framework of a critical overview which aims to indicate research perspectives. The contents mainly refer to modelling by fluid dynamic and kinetic equations and are arranged in three parts. The first part refers to methodological aspects of mathematical modelling and to the interpretation of experimental results. The second part is devoted to modelling and deals both with methodological aspects and with the description of some specific models. The third part reports about an overview on applications and research perspectives.