Speaker:

Reinhard Illner, University of Victoria

Title:

From Fokker-Planck type kinetic traffic models tostop-and-go waves in
dense traffic

Abstract:

We discuss kinetic models of Fokker-Planck type for multilane traffic
flow and compare them with models of conservation law type from
conceptual and practical points of view. The kinetic models allow
calculations of fundamental diagrams (density-flux diagrams) in
equilibrated traffic and offer in particular an explanation why such
diagrams appear to be multi-valued when lane changing is included. The
modeling suggests that lane-changing is necessary for this phenomenon
to occur, and allow to predict fluxes as functions of density with or
without lane changes.

If, in dense traffic, "diffusive' effects in driver behaviour becomes
small, the Fokker-Planck models degenerate into a Vlasov-type kinetic
equation with spatial nonlocality (nonlocality is a hallmark of all
these models). An ansatz $f(x,v,t)= \rho(x,t) \delta(v-u(x,t))$ leads
to macroscopic equations for $(\rho,u).$ Eliminating the nonlocality
by Taylor approximations leads to the pressureless gas dynamics
equations at the zeroth order, to PDEs of conservation type (more
precisely, of Hamilton-Jacobi type) like the Aw-Rascle model at the
first order, and to a system of equations of Hamilton-Jacobi equations
with diffusive corrections at second order. This latter case looks
complicated, but a search for traveling wave solutions produces
traveling waves that emulate the phenomenon of stop-and-go wave
formation on freeways. For each wave speed there appears to be a
velocity domain where traveling waves of that speed will not form
because the constant state $(\rho,u)$ is stable.

The latter work is a recent and ongoing collaboration with
M. Herty. The models were inspired by traffic observations made by
B. Kerner on the German autobahn, and our results are consistent with
these observations, at least from a qualitative point of view.