Phd in MATHEMATICAL MODELS FOR ENGINEERING, ELECTROMAGNETICS AND
NANOSCIENCES

Thesis title: Second order traffic flow models on road networks and real data applications

This thesis concerns macroscopic traffic models and data-driven models. In the first part we deal with the extension of Generic Second Order Models (GSOM) for traffic flow to road networks. We define a Riemann Solver at the junction based on a priority rule, providing an iterative algorithm able to build the solution to junctions with n incoming and m outgoing roads. The logic underlying our solver is the following: the flow is maximised respecting the priority rule, but the latter can be modified if the outgoing road supply exceeds the demand of the road with higher priority. We provide bounds on the total variation of waves interacting with the junction, giving explicit computations for intersections with two incoming and two outgoing roads. These estimates are fundamental to prove the existence of weak solutions to Cauchy problems on networks via Wave-Front-Tracking. GSOM are used to analyse traffic dynamics and their effects on the production of pollutant emissions. First we apply the proposed Riemann Solver to simulate traffic dynamics on diverge and merge junctions and on roundabouts obtained by combining these two types of intersection. Then, we propose a methodology to estimate the pollutant emissions deriving from traffic dynamics. The emission model is calibrated and validated using the NGSIM dataset of real trajectory data. Furthermore, we set up a minimisation problem aimed at finding the optimal priority rule for our Riemann Solver that reduces the emission rates due to the traffic dynamic. Finally, we analyse some chemical reactions which lead to the production of ozone, focusing on the effects on pollution of the presence of traffic lights on the road. Next, we introduce a macroscopic two-dimensional multi-class traffic model on a single road, aimed at including lane-changes and different types of vehicles. The multi-class model consists of a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems, we present a Lax-Friedrichs type discretisation scheme and we recover the theoretical results by means of numerical tests. We then calibrate and validate the multi-class model with real trajectory data and we test its ability of simulating vehicles overtaking. Finally, we present a new methodology to recover mass movements from snapshots of its distribution. To this end we put in place an algorithm based on the combination of two methods: first, we use the dynamic mode decomposition to create a system of equations describing the mass transfer; second, we use the Wasserstein distance to reconstruct the underlying velocity field that is responsible for the displacement. We conclude this part with a real-life application: the algorithm is employed to study the travel flows of people in large populated areas using, as input, presence data of people in given region domains derived from the mobile phone network, at different time instants.