Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models

TL;DR

This paper demonstrates that multi-valued fundamental diagrams in traffic flow can be systematically derived from jamiton solutions in classical second order models, revealing intrinsic traffic phase transitions.

Contribution

It shows that jamiton solutions in second order traffic models naturally produce set-valued fundamental diagrams, linking traffic phases to nonlinear wave solutions.

Findings

Set-valued fundamental diagrams are derived from jamiton solutions.

Transitions from function-valued to set-valued diagrams are natural in these models.

Models intrinsically reproduce traffic phase behavior.

Abstract

Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.