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Microscopic Selection of Solutions to Scalar Conservation Laws with Discontinuous Flux in the Context of Vehicular Traffic

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Semigroups of Operators – Theory and Applications (SOTA 2018)

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Abstract

In the context of road traffic modeling we consider a scalar hyperbolic conservation law with the flux (fundamental diagram) which is discontinuous at \(x=0\), featuring variable velocity limitation. The flow maximization criterion for selection of a unique admissible weak solution is generally admitted in the literature, however justification for its use can be traced back to the irrelevant vanishing viscosity approximation. We seek to assess the use of this criterion on the basis of modeling proper to the traffic context. We start from a first order microscopic follow-the-leader (FTL) model deduced from basic interaction rules between cars. We run numerical simulations of FTL model with large number of agents on truncated Riemann data, and observe convergence to the flow-maximizing Riemann solver. As an obstacle towards rigorous convergence analysis, we point out the lack of order-preservation of the FTL semigroup.

MDR is member of GNAMPA. MDR acknowledges the support of the National Science Centre, Poland, Project “Mathematics of multi-scale approaches in life and social sciences” No. 2017/25/B/ST1/00051, by the INdAM-GNAMPA Project 2019 “Equazioni alle derivate parziali di tipo iperbolico o non locale ed applicazioni” and by University of Ferrara, FIR Project 2019 “Leggi di conservazione di tipo iperbolico: teoria ed applicazioni”.

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Acknowledgements

The publication has been prepared with the support of the RUDN University Program 5-100.

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Authors and Affiliations

  1. Institut Denis Poisson (CNRS UMR7013), Université de Tours, Université d’Orléans, 37200, Tours, Parc Grandmont, France

    Boris Andreianov

  2. Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

    Boris Andreianov

  3. Department of Mathematics and Computer Science, University of Ferrara, I-44121, Ferrara, Italy

    Massimiliano D. Rosini

  4. Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, pl. Marii Curie-Skłodowskiej 1, 20-031, Lublin, Poland

    Massimiliano D. Rosini

Authors
  1. Boris Andreianov
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Correspondence to Massimiliano D. Rosini .

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Editors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, South Africa

    Jacek Banasiak

  2. Department of Mathematics, Lublin University of Technology, Lublin, Poland

    Adam Bobrowski

  3. Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland

    Mirosław Lachowicz

  4. Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

    Yuri Tomilov

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Andreianov, B., Rosini, M.D. (2020). Microscopic Selection of Solutions to Scalar Conservation Laws with Discontinuous Flux in the Context of Vehicular Traffic. In: Banasiak, J., Bobrowski, A., Lachowicz, M., Tomilov, Y. (eds) Semigroups of Operators – Theory and Applications. SOTA 2018. Springer Proceedings in Mathematics & Statistics, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-030-46079-2_7

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