On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models

Helbing, Dirk; Johansson, Anders · 2009 · OpenAlex-citations

DOI: 10.1140/epjb/e2009-00182-7

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Summary

This paper addresses the theoretical controversy surrounding second-order macroscopic traffic flow models, specifically focusing on Carlos Daganzo’s criticisms and Aw and Rascle’s subsequent attempts to resolve them. Daganzo argued that second-order models, which include dynamic velocity equations, suffer from theoretical inconsistencies, most notably the prediction of characteristic speeds faster than the average vehicle velocity. He claimed this violates causality by implying that future traffic conditions are influenced by events occurring behind a vehicle. The authors revisit these arguments to determine whether such faster characteristic speeds represent genuine physical errors or artifacts of model approximations. The study employs linear stability analysis to examine the propagation of perturbations in traffic flow. The authors derive general formulas for the stability thresholds and characteristic speeds of macroscopic models that account for velocity-dependent optimal velocities and traffic pressure. They compare these macroscopic results with microscopic traffic models to assess consistency. The analysis distinguishes between characteristic speeds, phase velocities, and group velocities, deriving conditions under which these propagation speeds exceed the average vehicle speed. The authors specifically analyze the Payne-Whitham model and the Aw-Rascle model, the latter of which was designed to ensure one characteristic speed equals the vehicle speed while the other is slower. The findings reveal that most macroscopic models with dynamic velocity equations predict two characteristic speeds, one of which is indeed faster than the average velocity. However, the authors argue that this does not necessarily constitute a violation of causality. They identify several potential explanations for this phenomenon, including artifacts from gradient expansions in approximations, indirect long-range interactions on circular roads, and the specific definition of perturbation propagation speeds. By comparing macroscopic models with microscopic counterparts, the paper demonstrates that microscopic models also exhibit similar propagation characteristics, suggesting that the "faster" characteristic speed may reflect realistic vehicle interactions rather than a theoretical flaw. The analysis further shows that incorporating velocity-dependent traffic pressure and optimal velocity functions can stabilize traffic flow and resolve issues such as negative vehicle velocities. The significance of this work lies in clarifying the physical interpretation of characteristic speeds in traffic modeling. The authors conclude that Daganzo’s criticism, while highlighting important deficiencies in earlier second-order models, may have misinterpreted the nature of characteristic speeds. The paper suggests that the presence of a characteristic speed greater than the vehicle speed is not inherently inconsistent if properly understood within the context of vehicle interactions and model derivations. This contributes to the ongoing development of more accurate macroscopic traffic models by providing a rigorous framework for evaluating their stability and physical validity.

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