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Convergence of a misanthrope process to the entropy solution of 1D problems

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  • Eymard, R.
  • Roussignol, M.
  • Tordeux, A.

Abstract

We prove the convergence, in some strong sense, of a Markov process called “a misanthrope process” to the entropy weak solution of a one-dimensional scalar nonlinear hyperbolic equation. Such a process may be used for the simulation of traffic flows. The convergence proof relies on the uniqueness of entropy Young measure solutions to the nonlinear hyperbolic equation, which holds for both the bounded and the unbounded cases. In the unbounded case, we also prove an error estimate. Finally, numerical results show how this convergence result may be understood in practical cases.

Suggested Citation

  • Eymard, R. & Roussignol, M. & Tordeux, A., 2012. "Convergence of a misanthrope process to the entropy solution of 1D problems," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3648-3679.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:11:p:3648-3679
    DOI: 10.1016/j.spa.2012.07.002
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    References listed on IDEAS

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    1. Paul I. Richards, 1956. "Shock Waves on the Highway," Operations Research, INFORMS, vol. 4(1), pages 42-51, February.
    2. Daganzo, Carlos F., 1994. "The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory," Transportation Research Part B: Methodological, Elsevier, vol. 28(4), pages 269-287, August.
    3. Sumalee, A. & Zhong, R.X. & Pan, T.L. & Szeto, W.Y., 2011. "Stochastic cell transmission model (SCTM): A stochastic dynamic traffic model for traffic state surveillance and assignment," Transportation Research Part B: Methodological, Elsevier, vol. 45(3), pages 507-533, March.
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