------------------------------------------ This talk will give an introduction to the dynamics of car-following models of highway traffic, which model vehicles as discrete entities moving in continuous space and time. The focus and tone will be qualitative analysis of simplified mathematical models of single-lane traffic, rather than complex engineering models which attempt the faithful capture of all behavioural aspects.

The typical structure is that the nth vehicle's displacement-velocity pair (x_n,v_n) satisfies the ordinary differential equations

dot{x}_n = v_n, dot{v}_n = f(h_n, dot{h}_n, v_n),

which couple the motion to the vehicle in front via the headway variable h_n = x_{n-1}-x_n. Extensions that will be discussed include "multi-anticipation" (coupling to several vehicles ahead) and reaction-time delay (giving rise to delay differential equations).

Under mild and plausible conditions on f, I will show that such systems admit a one-parameter family of solutions known as "uniform flows" and I will then show how to investigate their linear stability. It may be shown that if linear stability is lost, then it happens via the same generic mechanism and this tends to give rise to solutions which resemble (empirically observed) stop-and-go waves.

There are now 3 high level points to discuss:

1. Unfortunately, the (linear) dispersion relation of car-following models is in general incompatible with that of the GSOM theory (Aw-Rascle, Zhang, Lebacque) based on hyperbolic PDEs. Moreover, the GSOM theory, although it has many desirable properties built into it, cannot capture spontaneous jam formation nor can it model the downstream interface of a stop-and-go wave correctly. (My assertions about stop-and-go wave structure will be supported by some very recent empirical data from the UK's M42 motorway, which I believe is one of the most densely instrumented in the world.) So I believe that the GSOM structure, for all its merit, needs some refinement / relaxation.

2. Much has been made in previous meetings of the propensity of the Bando OV model to have crashes. I believe this is a distraction. It is rather simple to illustrate modifications, as used in commercial engineering simulations, which tend to prevent collisions. But it is very hard to prove general theorems about this. I should say that such modifications operate at a fully-nonlinear level (i.e. far from the uniform flow equilibria) and hence do not change any of my general conclusions regarding the linear dispersion relation.

3. Recent simulation work by Treiber, Kesting and Helbing has proposed that the distinction between linear and nonlinear stability is key in explaining complex flow patterns in empirical data. I agree with this view and I believe some of the conflicts described above can be resolved via the concept of nonlinear instability. Specifically: I will display car-following models with linearly stable equilibria yet with coexisting large ampliture stop-and-go wave solutions.

My hope is that this workshop will start a general methodology which derives PDE models from car-following ones, where the linear dispersion relation is inherited correctly, and yet where at large amplitude one somehow "snaps" into strictly hyperbolic behaviour, with all its decent properties. This goal seems to me somewhat beyond us at the moment.

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

In this mathematically oriented mini-course lecture, I will discuss the following topics.

First, I will recall the basic theory of the class of "second order" models of traffic flow introduced years ago with Aw (AR), and its main hyperbolic ingredients (Riemann problem, Godunov scheme ...). Let us recall (Lebacque et al, 2007) that one of the phase transitions models of Colombo turns out to be a variant of this class of models.

I will then describe the Lagrangian version, and I will show how one can make a rigorous link with Microscopic Follow the Leader Models of the form:

dx_i /dt = v_i ,

dv_i /dt = F (x_{i+1} - x_i) . (v_{i+1} - v_i) + A V(x_{i+1} - x_i) - v_i)

(with suitable assumptions on functions F and V) via a very natural "hyperbolic" scaling: (x',t') := (eps x , eps t), eps ---> 0, at least in the purely hyperbolic case: A=0, or, see below, under the Whitham stability subcharacteristic condition (SC).

I will then focus on the issue of the existence (?) of invariant regions for such systems: I will recall the celebrated result of Chueh-Conley-Smoller (Indiana Univ Math. J., 1977), which gives necessary and sufficient conditions for the existence of such regions. In particular, the gas dynamics system does not preserve the sign of the velocity. Similarly, already at the discrete level, Bando's optimal velocity model can present crashes.

Finally, the subcharacteristic condition (SC) states that for stability when eps ---> 0, the first part ("convective") must dominate the second part ("relaxation" ) of the above discrete model. If time allows, I will explain what happens under this condition.

COMMENTS: this mini-course lecture is intended to introduce the discussion, not only for J. Greenberg's lecture but also for the round tables. The underlying view is that any reasonable model should not be inconsistent at any scale, in particular at small scales: what's the sense of a subtle discussion on the dynamics at large time, if the model does not guarantee that the drivers will still be alive at the end of the trip? I think that this constraint practically forces the convective part of the system to look more or less like this AR system! On the other hand, in order to have a richer dynamics, contrarily to the (too much) clean situation where condition (SC) is satisfied, there is definitely a need for a partial violation of (SC), however still undergoing neither crashes nor negative velocities ... This issue of competition/complementarity between the hyperbolic part and the relaxation term at various scales should be a nice subject for our round tables!

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

In my lecture I'll follow on where Michel leaves off and attempt to cover the following topics in some detail.

I'll start with the Aw-Rascle-Greenberg model and show that both the discrete and continuum versions of this model have no velocity reversals and no crashes provided the term giving rise to the anticipatory acceleration is large enough. This result is basically the “ invariant region “ result alluded to by M. Rascle. The details of this Theorem will be presented and, contrary to what is commonly thought, this result is not (the same as)/(equivalent to) the Whitham sub-characteristic condition.

I will also look in some detail at what obtains for the discrete and continuum versions of the Bando model. There is a certain inconsistency between the discrete and continuum versions of the Bando model but there is a fix for the Bando continuum model which reconciles this inconsistency. In this lecture I'll show how the imposition of this fix transforms the continuum version of Bando into the much studied Payne-Whitham hydrodynamic model.

None of the Bando and related models have an invariant region like the ARG models and all give rise to crash scenarios and velocity reversals.

If time permits, I'll discuss large amplitude stop and go waves for both the ARG and Bando models.

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

In the first part, I will summarize several "stylized facts" of traffic flow, i.e., qualitative, and sometimes quantitative, dynamical aspects that has been observed in many countries and in various situations. I will distinguish between local phenomena such as the wide scattering of flow-density data observed at certain road cross sections in case of congested traffic, and spatiotemporal phenomena such as different types of congested traffic. All facts are backed up by data from several freeways in Germany, Holland, and the USA.

In the second part, I will present attempts to capture these phenomena by microscopic (and also macroscopic) models. In particular, there are two approaches to describe the spatiotemporal phenomena, which are considered to be mutually inconsistent: The "three-phase theory" of Boris Kerner, and the alternative concept of a "dynamic phase diagram" of congested states. I will show that some (but not all) discrepancies between the two approaches result just by looking differently at the same data. To illustrate this point, I will simulate aspects of the three-phase theory with several microscopic (and macroscopic) models that are deemed to be inconsistent with this theory. Conversely, all five phases of the ``phase diagram'' could be reproduced with models designed for the three-phase theory. It seems that models from three-phase theory do not explain more (or less) of the observed traffic phenomena compared to some of the simpler conventional models. Can we therefore apply Occam's razor?

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

We will consider a continuum traffic model with built-in phase transitions from free to congested traffic (respectively modeled by second order and first order models). The phase transition is obtained through the saturation of a maximal density constraint which depends on the average flow velocity. The model is obtained through a singular limit from the Aw-Rascle traffic model. An important feature of this model is that its stationary states spontaneously obey the fundamental diagram, without the need for any additional relaxation term, by contrast to standard second order models. This in particular shows that the fundamental diagram is built inside the microscopic kinetics of the particles.

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

We study very simple microscopic models of Bando-type. The aim is to understand the full dynamics of such simple models.
We recover macroscopic properties like lambda-shaped fundamental diagramms.
The mathematical approach is general and easily applicable to the cases of inhomogeneous drivers or roadworks.

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

This talk will overview some properties of a hyperbolic model for traffic flow based on phase transitions.

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)

• Event: Workshop and Minicourse on 'Multiscale Problems and Models in Traffic Flow' (2008)