Nonlinear coupled systems of PDEs for modeling of multi-lane traffic flow problems
Keywords: Engineering.

In this dissertation, we develop a traffic flow model that provides general behavioral insights of congestion on traffic corridors. The computational tool is built in different stages in which more complex scenarios can be studied. First, we extend the traditional Lighthill-Whitham-Richards (LWR) model, based on partial differential equations (PDEs), for unidirectional traffic on a single road. This scalar model PDE-based traffic flow simulation uses a linear velocity model. To create PDE-based traffic flow models that take into account more realistic driver behaviors, it has been proposed to extend the scalar model to a system of two PDEs, inspired by fluid dynamics, such that the added equation, satisfied by the velocity function, mimics the momentum equation. Two classical second-order models are the Payne-Whitham (PW) and the Aw-Rascle (AR) models. We choose to instead explore the possibility to design non-linear velocity models in the scalar PDE to capture desirable behavior while retaining the satisfaction of the L1 contractivity of the LWR model. We develop a discrete model, in which cars are represented by particles, to inform the choice of velocity functions for the PDE model. We derive desirable mathematical conditions for velocity functions to ensure the PDE's well-posedness. We incorporate various driver behaviors in the particle-based model generating a family of velocity functions. We show, for a variety of traffic scenarios, that the PDE solutions map back to the traffic distributions on the road obtained with the discrete models. We explore the impacts of changes in average and individual driving behaviors on the velocity profiles and the PDE solutions. Second, we generalize the initial one-lane framework to a multi-lane setting, based on the model developed by Colombo and Corli, in which we incorporate realistic driver behaviors through a family of non-linear source functions. We derive desirable mathematical conditions for the source functions to ensure L1 contractivity for the system of PDEs. We extend the particle-based traffic model to illustrate multi-lane roads and inform the choice of source functions for the PDE model. We add various driver behaviors, associated with lane switching, in the discrete model to generate realistic source functions. We also explore the impacts of variations in the newly incorporated driving behaviors on different traffic scenarios, at the individual scale and average driver tendencies, and analyze how the source profiles generated by the discrete model change. Third, building on the multi-lane model, we study converging and diverging lanes and evaluate their impact on traffic. We extend the discrete model to inform the choice of source and velocity functions and include a variety of driver behaviors associated with moving to the temporary lane. We explore the effect of variations in driver behaviors on the source and velocity profiles generated by the discrete model. We derive optimal implementations of converging and diverging lanes, in terms of improving traffic efficiency and mitigating traffic jams, for Riemann problems. We consider changes in the initial car density on the road and incoming flux of car density, addition of carpool or toll lanes and temporary closure of a lane.