Kinetic description of bottleneck effects in traffic flow

Abstract

This paper deals with the effects of traffic bottlenecks using an extended Lighthill-Whitham-Richards (LWR) model. The solution structure is analytically indicated by the study of the Riemann problem characterized by a discontinuous flux. This leads to a typical solution describing a queue upstream of the bottleneck and its width and height, and informs the design of a δ-mapping algorithm. More significantly, it is found that the kinetic model is able to reproduce stop-and-go waves for a triangular fundamental diagram. Some simulation examples, which are in agreement with the analytical solutions, are given to support these conclusions.

Steady-state traffic flow on a ring road with up- and down-slopes

“…The constancy of the flow at the dividing point is attributed to the mass conservation or the Rankine-Hugoniot condition (see [2][3][4][5] and the references therein). Let ρ = ρ ∓ and Q ∓ e denote the densities and fundamental diagrams on the left and right hand sides of the interface.…”

“…For the dividing point which is stationary, a "wave entropy" condition can be applied at the interface to describe valid wave breaking on the both sides [2][3][4][5]. That is, the characteristic speeds on the two sides take the same sign with…”

“…Actually, such a steady-state solution could be described by using the theory of hyperbolic conservation laws with discontinuous fluxes [5,12], which is exact and stable for the first-order continuum LWR model [15,16]. However, it is only an approximate solution to the car-following model or the higher-order model with relaxation if the relaxation time is sufficiently small; otherwise, it is unnecessarily stable due to the underlying metastability in these models.…”

“…Traffic flow problem with inhomogeneous road conditions has been studied by using the macroscopic continuum model and the microscopic car-following model. In the former formulation, the Riemann problem was concerned mainly for designing a numerical scheme, which could be related to the theory of hyperbolic conservation laws with discontinuous fluxes [1][2][3][4][5][6][7][8]. In the latter formulation, steady-state flow was analytically or numerically studied by assuming or implying that the solution is piecewise equilibrium constant in the inhomogeneous road sections [9][10][11][12][13][14].…”