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On the Controversy Around Daganzo’s Requiem for and Aw–Rascle’s Resurrection of Second-Order Traffic Flow Models

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Modelling and Optimisation of Flows on Networks

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2062))

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Abstract

Daganzo’s criticisms of second-order fluid approximations of traffic flow [C. Daganzo, Transp. Res. B 29, 277–286 (1995)] and Aw and Rascle’s proposal how to overcome them [A. Aw and M. Rascle, SIAM J. Appl. Math. 60, 916–938 (2000)] have stimulated an intensive scientific activity in the field of traffic modeling. Here, we will revisit their arguments and the interpretations behind them. We will start by analyzing the linear stability of traffic models, which is a widely established approach to study the ability of traffic models to describe emergent traffic jams. Besides deriving a collection of useful formulas for stability analyses, the main attention is put on the characteristic speeds, which are related to the group velocities of the linearized model equations. Most macroscopic traffic models with a dynamic velocity equation appear to predict two characteristic speeds, one of which is faster than the average velocity. This has been claimed to constitute a theoretical inconsistency. We will carefully discuss arguments for and against this view. In particular, we will shed some new light on the problem by comparing Payne’s macroscopic traffic model with the Aw–Rascle model and macroscopic with microscopic traffic models.

First published in: The European Physical Journal B 69(4), 549–562, DOI: 10.1140/epjb/e2009-00182-7 (2009), © EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2009, reproduction with kind permission of The European Physical Journal (EPJ).

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Notes

  1. 1.

    A typical example is the modulation of electromagnetic waves used to transfer information via radio.

  2. 2.

    Note that the existence of perturbations in the traffic flow always implies a variation of the vehicle speeds.

  3. 3.

    Simulations for open boundary conditions basically yield the same results as for periodic boundary conditions, given the system (in terms of the road length L) is sufficiently large.

  4. 4.

    Of course, this does not speak against models of the Aw–Rascle type.

  5. 5.

    The idea behind the characteristics is to introduce a parameterization \(t({s}_{1},{s}_{2})\), \(x({s}_{1},{s}_{2})\), which is defined by \(\partial t/\partial {s}_{j} = 1\) and \(\partial x/\partial {s}_{j} = {C}_{j}\). Then, one can rewrite (68) as \(\frac{\partial {y}_{j}} {\partial {s}_{j}} = \frac{\partial {y}_{j}(x,t)} {\partial t} \, \frac{\partial t} {\partial {s}_{j}} + \frac{\partial {y}_{j}(x,t)} {\partial x} \, \frac{\partial x} {\partial {s}_{j}} = {({\underline{R}}^{-1}\underline{B}\,\underline{R}\,\mathbf{y})}_{j}\,.\) In the generalized coordinates s 1 and s 2, the partial differential equations in x and t we were starting with, turn into ordinary differential equations. These are much easier to solve.

  6. 6.

    Note that formulas (69) and (70) only apply to the limiting case \(\tau \rightarrow \infty \), where the relaxation term of the macroscopic traffic model vanishes.

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Acknowledgements

Author contributions: DH performed the analytical calculations and proposed the initial conditions for the simulation presented in Fig. 1. AJ generated the computational results and prepared the figure.

Acknowledgment: DH would like to thank for the inspiring discussions with the participants of the Workshop on “Multiscale Problems and Models in Traffic Flow” organized by Michel Rascle and Christian Schmeiser at the Wolfgang Pauli Institute in Vienna from May 5–9, 2008, with partial support by the CNRS.

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  1. ETH Zurich, UNO D11, Universitätstr. 41, 8092, Zurich, Switzerland

    Dirk Helbing & Anders Johansson

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Correspondence to Dirk Helbing .

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Helbing, D., Johansson, A. (2013). On the Controversy Around Daganzo’s Requiem for and Aw–Rascle’s Resurrection of Second-Order Traffic Flow Models. In: Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics(), vol 2062. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32160-3_4

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