Non-Equilibrium Dynamics of Second Order Traffic Models

Abstract

Even though first order LWR models have many limitations, they are still widely used in many engineering applications. Second-order models, on the other hand, address many of those limitations. Among second-order models, the inhomogeneous Aw-Rascle-Zhang (ARZ) model is well-received as its structure generates characteristic waves that make physical sense. The ARZ model --- and other $2\times 2$ hyperbolic systems with a relaxation term --- possess a critical phase transition: whenever the sub-characteristic condition (SCC) is violated, uniform traffic flow is unstable, and small perturbations grow into nonlinear traveling waves, called jamitons. The case where the SCC is satisfied has been studied extensively. However, what is essentially unstudied is the question: which jamiton solutions are dynamically stable? To understand which stop-and-go traffic waves can arise through the dynamics of the model, this question is critical. This dissertation first outlines the mathematical foundations of the ARZ model and its solutions, then presents a computational study demonstrating which types of jamitons are dynamically stable, and which are not. After that, a procedure is presented that characterizes the stability of jamitons. The study reveals that a critical component of this analysis is the proper treatment of the perturbations to the shocks, and of the neighborhood of the sonic points. The insight gained from answering the question regarding the dynamical stability of jamitons has many applications. One particular application presented here is deriving an averaged model for the ARZ model. Such a model is as simple to solve (analytically and numerically) as the LWR model, but nevertheless captures the cumulative effects of jamitons regarding fuel consumption, total flow, and braking events.