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The Aw-Rascle and Zhang's model: Vacuum problems, existence and regularity of the solutions of the Riemann problem

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  • Lebacque, Jean-Patrick
  • Mammar, Salim
  • Haj-Salem, Habib

Abstract

In this paper we will discuss some aspects of the recent macroscopic models of the second-order proposed by [Aw, A., Rascle, M., 2000. Resurrection of second order models of traffic flow. SIAM Journal of Applied Mathematics 60 (3), 916-938] and [Zhang, H.M., 2002. A non-equilibrium traffic model devoid of gas-like behavior. Transportation Research Part B 36, 275-290]. These models were suggested after the publication of an article written by [Daganzo, C.F., 1995. Requiem for second-order fluid approximations of traffic flow. Transportation Research Part B 29, 277-286] showing that some classical second-order models can exhibit non-physical solutions. It is shown in this note that the ARZ (Aw-Rascle-Zhang) model respects the anisotropic character of traffic flow, that it yields physical solutions, and that vacuum problems can be solved satisfactorily, provided that the fundamental diagram (equilibrium speed-density relationship) is extended in a suitable fashion. It follows that the Riemann problem for the ARZ model with extended fundamental diagram always admits a solution, and that this solution depends continuously on the initial conditions.

Suggested Citation

  • Lebacque, Jean-Patrick & Mammar, Salim & Haj-Salem, Habib, 2007. "The Aw-Rascle and Zhang's model: Vacuum problems, existence and regularity of the solutions of the Riemann problem," Transportation Research Part B: Methodological, Elsevier, vol. 41(7), pages 710-721, August.
    • Handle: RePEc:eee:transb:v:41:y:2007:i:7:p:710-721
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    References listed on IDEAS

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    1. Ross, Paul, 1988. "Traffic dynamics," Transportation Research Part B: Methodological, Elsevier, vol. 22(6), pages 421-435, December.
    2. Zhang, H. M., 2002. "A non-equilibrium traffic model devoid of gas-like behavior," Transportation Research Part B: Methodological, Elsevier, vol. 36(3), pages 275-290, March.
    3. Daganzo, Carlos F., 1995. "Requiem for second-order fluid approximations of traffic flow," Transportation Research Part B: Methodological, Elsevier, vol. 29(4), pages 277-286, August.
    4. Zhang, H. M., 1998. "A theory of nonequilibrium traffic flow," Transportation Research Part B: Methodological, Elsevier, vol. 32(7), pages 485-498, September.
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