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 non-equilibrium 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.
Highlights
There are mainly three modeling scales in the mathematical description of vehicular traffic flow
We study spatially non-homogeneous kinetic models for vehicular traffic flow
Using a Chapman-Enskog expansion we show that the BGK equation for traffic flow problems yields an advection-diffusion equation having a negative diffusion coefficient in dense traffic
Summary
There are mainly three modeling scales in the mathematical description of vehicular traffic flow. This analysis is closely related to the investigation of the sub-characteristic condition for relaxation systems [6, 23] This implies possible unbounded growth of non-equilibrium waves and the corresponding behavior is investigated at the particle description and at the macroscopic level of the BGK model by looking at the system of the second-order moment equations. This approach has the advantage of prescribing the mesoscopic step between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model. The maps ρ → Qeq(ρ) and ρ → Ueq(ρ) are functions prescribing a uniquely defined correspondence between the density and the quantities (7) These relations provide the so-called simulated fundamental diagrams of traffic which are used in order to validate the model.
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