A Wave-Based Resolution Scheme for the Hydrodynamic LWR Traffic Flow Model

Henn, Vincent

doi:10.1007/3-540-28091-x_10

Cited by 15 publications

(12 citation statements)

References 19 publications

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“…The maximum free flow and maximum congested flow are q m = q mc = 1800 veh/h; update time T s = 1 s; the inflow vector of onramp is r 0 = 360 veh/h and r 1 = 0 ; the upstream flow q 2 = 1440 veh/h, and downstream flow q 0 = 1800 veh/h. Therefore, the uncongested equilibrium state [20,16] T and the most congested equilibrium …”

Section: A the Ctms And Lctms With A Trapeziform-shape Fundamental Dmentioning

confidence: 99%

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Equilibrium analysis and comparison for general CTMs and LCTMs

Chen

Shi

et al. 2010

13th International IEEE Conference on Intelligent Transportation Systems

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In this paper, we study the equilibriums of the general Cell transmission models (CTM) and the general Lagged Cell transmission models (LCTM). Theoretical analyses show that different CTMs and LCTMs converge to a two-segment almost piecewise-linear equilibrium state with similar shapes, although the variety of the traffic flow-density relationships (fundamental diagram, FD) and the existence of the lagged effect lead to different patterns of convergence trajectories and speeds. These results indicate the usefulness of the CTMs and LTCMs in practice, especially when we are interested on the formations and dissipations of traffic congestions.

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Section: A the Ctms And Lctms With A Trapeziform-shape Fundamental Dmentioning

confidence: 99%

“…The evolution of traffic density within each cell of freeway has been explored as the rarefaction wave under the Riemann initial condition [20]. Each demand pattern may induce a unique equilibrium density vector and the infinite set of equilibrium states characterized by the stability of traffic flow.…”

Section: Equilibrium Analysis Of the General Ctms And Lctmsmentioning

confidence: 99%

Equilibrium analysis and comparison for general CTMs and LCTMs

Chen

Shi

et al. 2010

13th International IEEE Conference on Intelligent Transportation Systems

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“…In this paper we approach this challenge by looking for an exact non-entropic solution to the kinematic wave model. As argued in Henn (2003), such solutions may be easier to calculate, although there will be some loss in physical relevance. Two non-entropic solutions in particular are of interest, namely solutions that are consistent with shockwave theory (also referred to as shockwave analysis, see e.g., May, 1990), and solutions that are based on a scheme as proposed in the front tracking algorithm in Henn (2003).…”

Section: Assumptions and Aim Of The Studymentioning

confidence: 99%

“…As argued in Henn (2003), such solutions may be easier to calculate, although there will be some loss in physical relevance. Two non-entropic solutions in particular are of interest, namely solutions that are consistent with shockwave theory (also referred to as shockwave analysis, see e.g., May, 1990), and solutions that are based on a scheme as proposed in the front tracking algorithm in Henn (2003). Both approaches only simplify expansion fans, such that the solution is in fact entropic when flows do not increase, and is approximate entropic when flows increase.…”

Section: Assumptions and Aim Of The Studymentioning

confidence: 99%

Continuous-time general link transmission model with simplified fanning, Part I: Theory and link model formulation

Bliemer

Raadsen

2019

Transportation Research Part B: Methodological

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The kinematic wave theory is widely used to simulate traffic flows on road segments. Link transmission models are methods to find a solution to the kinematic wave model, however, their computational efficiency heavily relies on the shape of the fundamental diagram that is used as input. Despite the limitations and drawbacks of triangular and piecewise linear fundamental diagrams, they remain popular because they result in highly efficient algorithms. Using smooth nonlinear branches is often preferred in terms of realism and other desirable properties, but this comes at a significantly higher computational cost and requires time discretisation to find an approximate solution. In this paper we consider a nonlinear fundamental diagram as input and propose on-the-fly multi-step linearization techniques to simplify expansion fans. This leads to two simplified link transmission models that can be solved exactly in continuous time under the assumption of piecewise stationary travel demand. One of the models simplifies to shockwave theory in case of a single step. We show that embedding shockwave theory in the link transmission model allows for finding an exact solution in continuous time and we discuss the potential for the design of efficient event-based algorithms for general networks.

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“…• the wave tracking method [21,25] which simplifies the calculations by only following the meaningful kinematic waves that is shocks or rarefaction waves. However this method is event-based since it needs to evaluate each time when two kinematic waves meet and in practice it is restricted to piecewise linear flow-density fundamental diagrams (and piecewise initial and boundary conditions) for dealing with a finite number of wave speeds.…”

Section: Computational Aspects For the Lwr Modelmentioning

confidence: 99%

A variational formulation for higher order macroscopic traffic flow models: Numerical investigation

Costeseque

Lebacque²

2014

Transportation Research Part B: Methodological

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This paper deals with numerical methods providing semi-analytic solutions to a wide class of macroscopic traffic flow models for piecewise affine initial and boundary conditions. In a very recent paper, a variational principle has been proved for models of the Generic Second Order Modeling (GSOM) family, yielding an adequate framework for effective numerical methods. Any model of the GSOM family can be recast into its Lagrangian form as a Hamilton-Jacobi equation (HJ) for which the solution is interpreted as the position of vehicles. This solution can be computed thanks to Lax-Hopf like formulas and a generalization of the inf-morphism property. The efficiency of this computational method is illustrated through a numerical example and finally a discussion about future developments is provided.Keywords: Traffic flow, Hamilton-Jacobi equation, Lax-Hopf algorithm, Lagrangian. Introduction General backgroundMacroscopic traffic flow modeling. In order to get a realistic estimation of the real-time traffic states on networks, traffic operators and managers need macroscopic traffic flow models. These models must be simple, robust, allowing to get solutions at a low computational cost. The main macroscopic models are based on conservation laws or hyperbolic systems (see [30,19] for a review). The seminal LWR model (for Lighthill-Whitham and Richards) was proposed in [42,51] as a single conservation law with unknown the vehicles density. This model based on a first order Partial Differential Equation (PDE) is very simple and robust but it fails to recapture some empirical features of traffic. In particular, it does not allow to take into account non-equilibrium traffic states mainly in congested situation. More sophisticated models referred to as higher order models were developed to encompass kinematic constraints of real vehicles or also the wide variety of driver behaviors, even at the macroscopic level. In this paper we deal with models of the Generic Second Order Modeling (GSOM) family. Even if these models are more complicated to deal with, they permit to reproduce traffic instabilities (such as the so-called stop-and-go waves, the hysteresis phenomenon or capacity drop) which move at the traffic speed and differ from kinematic waves [53] Traffic flow monitoring. Before the wide propagation of internet handsets, traffic monitoring has mainly been built on dedicated infrastructure which imply quite important installation and maintenance costs. Traffic flow monitoring and management has been deeply modified with the development of new technologies in mobile sensing aiming to provide a quite important quantity of floating car data. Traffic flow models are needed to be well suited such that managers could use both Eulerian and Lagrangian data for improving traffic state estimation. The term Eulerian refers to "classical" fixed equipment giving records of occupancy or flow of vehicles on a freeway section. This kind of measurements come from e.g. fixed inductive loop detectors, Radio Frequency Identification (R...

show abstract

A Wave-Based Resolution Scheme for the Hydrodynamic LWR Traffic Flow Model

Cited by 15 publications

References 19 publications

Equilibrium analysis and comparison for general CTMs and LCTMs

Equilibrium analysis and comparison for general CTMs and LCTMs

Continuous-time general link transmission model with simplified fanning, Part I: Theory and link model formulation

A variational formulation for higher order macroscopic traffic flow models: Numerical investigation

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