The BGK approximation of kinetic models for traffic

Abstract

We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by deriving a modified formulation of the BGK-type equation. The new kinetic model allows to reproduce conditionally stable nonequilibrium phenomena in traffic flow. In particular, stop and go waves appear as bounded backward propagating signals occurring in bounded regimes of the density where the model is unstable. The BGK-type model introduced here also offers the mesoscopic description between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model.MSC 90B20, 35Q20, 35Q70

From Kinetic to Macroscopic Models and Back

“…The presence of instabilities is investigated using a formal Chapman-Enskog expansion. For the sake of simplicity, the analysis is performed using the BGK approximation of the Boltzmann-type collision kernel (3), for constant but small values of ε. Unstable waves are present also in this linearized setting [11].…”

“…However, in that work the stabilization is unfortunately too strong and it implies that, for example, stop-and-go waves will not occur. Following the approach introduced in [11], we derive a weakly unstable BGK model modifying the design of the space of microscopic speeds. Further, we obtain by suitable limits from this mesoscopic representation a microscopic follow-the-leader [9] or Bando [3] model, and a macroscopic Aw-Rascle [2] and Zhang [21] type model.…”

“…Finally, in Sect. 4 we derive a modified version of the BGK model, as in [11], and analyze the stability in the case of interactions with over-braking. In Sect.…”

Properties of the LWR model with time delay

“…Starting from the natural idea of tracking every single vehicle, several microscopic models, based on the idea of Follow-the-Leader, grew-up for computing positions, velocities and accelerations of each car by means of systems of ordinary differential equations (ODEs) [1,8,19,21,44]. Other ways go from kinetic [28,34,45] to macroscopic fluid-dynamic and measures approaches [2,11,12,24,33,38], focusing on averaged quantities, such as the traffic density and the speed of the traffic flow, by means of systems of hyperbolic partial differential equations (PDEs), in particular conservation laws. In this way we loose the detailed level of vehicles' description, indeed they become indistinguishable from each other.…”

“…This means that a lot of models have been developed in the last years, i.e. [7,22,28,37,42], and also several real experiments took place, just see [41,47].…”