The Aw-Rascle-Zhang model with constraints

TL;DR

This paper extends the Aw-Rascle-Zhang traffic model to include constraints like slow vehicles and traffic lights, introducing new Riemann solvers, analyzing their invariant domains, and developing numerical methods to capture solutions.

Contribution

It introduces two Riemann solvers for the constrained Aw-Rascle-Zhang model, characterizes their invariant domains, and proves existence of solutions for the Cauchy problem with constraints.

Findings

The first Riemann solver conserves both cars and momentum.

The second Riemann solver conserves only cars.

Numerical methods successfully capture solutions for the first Riemann solver.

Abstract

The thesis deals with the Aw-Rascle-Zhang model for traffic. We have applied the model to describe the influence of a large and slow vehicle (a bus or a truck) on the traffic. The trajectory of the bus is given by an ODE. The model can also be applied to the case of a fixed constraint, like a traffic light or a toll gate. We define two different Riemann solvers: the first one conserves both the number of cars and the generalized momentum, while the second conserves only the number of cars. We characterize the invariant domains for these Riemann solvers. We study two numerical methods based on the Godunov method to capture the proposed solutions and we track the bus trajectory with a front-tracking technique. The first method is based on conservation and captures exactly the solution corresponding to the first Riemann solver. The second method is based on a non-uniform mesh. Both methods fail to capture the solutions corresponding to the second Riemann solver for general initial data. Finally, we prove the existence of solutions for the Cauchy problem for the second Riemann solver in the case of a fixed constraint, applying the wave-front tracking method.