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978-3-8439-4903-3, Reihe Mathematik

Jan Friedrich Traffic flow models with nonlocal velocity

233 Seiten, Dissertation Universität Mannheim (2021), Softcover, A5

In this thesis, we present so–called nonlocal traffic flow models which possess more information about surrounding traffic. Such models become more and more important due to the progress in autonomous driving. They are described through partial differential equations conserving the mass. We focus on models in which the nonlocality is included via an integral evaluation of the velocity in terms of a convolution product in a certain nonlocal range. Here, we introduce a base model which considers the mean downstream velocity and assumes that the underlying speed law remains the same on the whole road. Moreover, we extend this nonlocal traffic flow model by allowing for a spatial change in the speed law. In order to approximate solutions, we introduce a numerical scheme of upwind type. We derive several properties of this scheme which enable us to prove the existence of weak (entropy) solutions. Furthermore, we prove the uniqueness of solutions: for the base model, we obtain uniqueness of weak solutions and for the extended model, uniqueness of weak entropy solutions.

Then, we provide further extensions of the presented models. First, we extend the nonlocal traffic flow model to networks while keeping the basic idea of being driven by a mean downstream velocity. We propose necessary assumptions on the coupling of different roads to obtain a well defined network model. Further, we consider the modeling of a single road more closely and distinguish between several lanes. The coupling of these lanes is induced by a nonlocal source term. In particular, this nonlocality can differ from the one responsible for the transport and can depend on down- and upstream traffic. Finally, starting from a microscopic model and proving rigorously the macroscopic limit, we introduce a second order nonlocal traffic flow model which conserves the mass and momentum. For all extensions, we present suitable numerical approximations and use these to prove the existence of weak solutions.