In this mathematically oriented minicourse 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 ChuehConleySmoller (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 minicourse 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!
