Abstract:

Stop-and-go waves, also called phantom jams, are often observed in real traffic flows but can be produced neither by the classical Lighthill-Whitham-Richards (LWR) model nor by other known continuum traffic models. To capture stop-and-go waves, we add hysteresis to the LWR model. For the model we propose, all possible viscous waves are found, and necessary and sufficient conditions for their existence are provided. In particular, deceleration and acceleration shocks appear; the latter were never rigorously defined before, in spite of the fact that they were observed  in real traffic flows. Stop-and-go waves can be  constructed by a pair of deceleration and acceleration shocks that completes a hysteresis cycle, illustrating how hysteresis loops lead to stop-and-go waves. Furthermore, stop-and-go waves are not present where anticipation (i.e., negative hysteresis) loops exist. Riemann solutions are then found for all possible Riemann data.  As a consequence, it is explicitly shown that, in the phase region where hysteresis loops exist, a sufficient deviation in speed of a few vehicles in an otherwise uniform car platoon can generate stop-and-go waves. While a one-time small speed deviation decays, it is demonstrated numerically that frequent small perturbations of drivers' mood will generate stop-and-go waves,  confirming that a uniform speed car platoon is metastable in congested zone, agreeing observations of real traffic experiments. In the region where there are negative hysteresis loops, the speed variation decays.

(If audience are graduate students or senior undergrad.)

1. Conservation of mass equation. 

Convert Eulerian to Lagrangian coordinate.

Viscous shocks, and rarefaction waves, Lax entropy condition.

Solutions to Riemann problems.

2. LWR model for traffic. deceleration shocks and acceleration rarefaction waves.

Wave interactions.

LWR model dissipates speed oscillations and hence it does not generate traffic jams.

Numerical solutions of LWR model.

3 . Real traffic jams data.

Sugiyama experiment.

Agreement and disagreements between LWR and real traffic.

Past effort of fixing LWR:  other macroscopic models.

Still cannot produce stop-and-go waves.

4. Hysteresis, what is it?

Including hysteresis into LWR.

New model = LWR+hysteresis

Waves of the model for traffic involving hysteresis.

Stop-and-go solutions. This is what past models cannot do.

5. Example solutions simulating real traffic phenomena.

Numerical scheme for the model.

The model's wave tracking analysis and numerical simulation resembles Sugiyama experiment.

A mechanism for traffic jam formation, and simulation.

6. Dissipating traffic jams using controlled autonomous cars