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

Seibold, Benjamin; Flynn, M. R.; Kasimov, Aslan R.; Rosales, Rodolfo R. · 2013 · OpenAlex-citations

DOI: 10.3934/nhm.2013.8.745

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Summary

This paper addresses the mathematical construction of set-valued fundamental diagrams (FDs) in vehicular traffic flow, specifically focusing on the congested regime where a single vehicle density corresponds to multiple flow rates. The authors demonstrate that this multi-valued behavior arises naturally within classical second-order macroscopic traffic models, such as the Payne-Whitham (PW) and inhomogeneous Aw-Rascle-Zhang (ARZ) models, without requiring explicitly prescribed phase transitions. The study establishes a direct link between the instability of uniform traffic flow and the existence of "jamitons"—self-sustained, attracting traveling wave solutions with embedded shocks that model stop-and-go traffic phenomena. The methodology involves analyzing inviscid, single-lane macroscopic models described by hyperbolic balance laws with relaxation terms. The authors utilize Whitham’s theorem, which establishes the equivalence between the linear stability of uniform base states and the sub-characteristic condition (SCC). They derive specific SCC criteria for both the PW and ARZ models, showing that uniform flow is stable only when the characteristic velocity of the reduced first-order Lighthill-Whitham-Richards model falls between the two characteristic velocities of the second-order model. The paper then adapts the Zel’dovich-von Neumann-Döring theory of detonations to construct jamiton solutions using a traveling wave ansatz in Lagrangian variables. This approach identifies jamitons as the mathematical manifestation of traffic instabilities that occur when the SCC is violated. The main findings confirm that jamitons exist if and only if the corresponding uniform base state solution is unstable. The authors prove that jamiton-dominated traffic flow never possesses a higher effective flow rate than uniform flow of the same average density. By systematically constructing these jamiton solutions, the paper generates set-valued fundamental diagrams that qualitatively reproduce the spread observed in empirical sensor data, such as the RTMC dataset from Minnesota. The results show that transitions from function-valued to set-valued FDs correspond precisely to the density ranges where jamiton solutions dominate, effectively reproducing traffic phases like free flow, synchronized flow, and wide moving jams intrinsically within the model structure. The significance of this work lies in demonstrating that complex traffic phenomena, including phantom jams and multi-valued flow-density relationships, are inherent properties of well-known second-order models. This eliminates the need for ad-hoc phase transition conditions in modeling congested traffic. The findings provide a rigorous theoretical foundation for understanding how macroscopic models capture the dynamics of traffic instabilities and offer a systematic method for deriving fundamental diagrams that align with real-world observational data.

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• traffic density