Traffic Flow

To minimise congestion it is necessary to understand traffic behaviour, which has led to many traffic related studies with varied perspectives. One approach is the application of a macroscopic model to characterise vehicle headway. In particular, we examine an optimal-velocity (OV) model that describes car motion on a single lane road. By applying linear stability analysis to this OV system, a neutral stability line is identified that signifies the boundary between the metastable and unstable regimes.

The OV model reduces to a perturbed Korteweg-de Vries (KdV) equation within the metastable zone using nonlinear theory, which is the KdV equation with higher order correction terms. We apply modulation theory to this perturbed KdV equation to obtain a family of cnoidal waves that represent traffic congestion. They are of a similar form to the KdV soliton, except they are spatially periodic. These solutions are validated with numerical simulations. Their long-time behaviour is also numerically investigated, revealing these cnoidal waves do eventually dissolve.

Additionally, solutions corresponding to the unstable region are analysed, where the traffic OV model transforms into a perturbed modified Korteweg-de Vries (mKdV) equation. Again applying modulation theory, a family of steady travelling wave solutions are highlighted that satisfy periodic boundary constraints. These describe the stop/start motion representative of a traffic jam. Numerical simulations are used to confirm our analysis, as well as to study the long-time dynamics. As a result, the travelling waves are shown to be numerically stable.

Traffic Flow 1






















L.L. Hattam, KdV cnoidal waves in a traffic flow model with periodic boundaries (2016).

L.L. Hattam, Travelling wave solutions of the perturbed mKdV equation that represent traffic congestion (2016).

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