Mathematical models and methods for traffic flow and stop & go waves

In this thesis we are concerned with mathematical methods and models for traffic flow, with special emphasis to second-order effects like Stop & Go waves. To begin with, we investigate the sensitivity of the celebrated Lighthill-Whitham-Richards model on network to its parameters and to the network itself. The quantification of sensitivity is obtained by measuring the Wasserstein distance between two LWR solutions corresponding to different inputs. To this end, we propose a numerical method to approximate the Wasserstein distance between two density distributions defined on a network. After that, we present a new multi-scale method for reproducing traffic flow, which couples a first order macroscopic model with a second-order microscopic model, avoiding any interface or boundary conditions between them. The new multi-scale model is characterized by the fact that microscopic and macroscopic descriptions are not spatially or temporally separated. Furthermore, a delayed LWR model on networks is proposed in order to allow simple first-order models to describe complex second-order effects caused by bounded accelerations. A time delay term is introduced in the flux term and its impact is studied from the numerical point of view. Lastly, we focus on Stop & Go waves, a typical phenomenon of congested traffic flow. Real data are used to point out the main features of this phenomenon, then we investigate the possibility to reproduce it using new traffic models specifically conceived for this purpose