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Runge–Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

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Abstract

We propose a bound-preserving Runge–Kutta (RK) discontinuous Galerkin (DG) method as an efficient, effective and compact numerical approach for numerical simulation of traffic flow problems on networks, with arbitrary high order accuracy. Road networks are modeled by graphs, composed of a finite number of roads that meet at junctions. On each road, a scalar conservation law describes the dynamics, while coupling conditions are specified at junctions to define flow separation or convergence at the points where roads meet. We incorporate such coupling conditions in the RK DG framework, and apply an arbitrary high order bound preserving limiter to the RK DG method to preserve the physical bounds on the network solutions (car density). We showcase the proposed algorithm on several benchmark test cases from the literature, as well as several new challenging examples with rich solution structures. Modeling and simulation of Cauchy problems for traffic flows on networks is notorious for lack of uniqueness or (Lipschitz) continuous dependence. The discontinuous Galerkin method proposed here deals elegantly with these problems, and is perhaps the only realistic and efficient high-order method for network problems.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77004, USA

    Suncica Canic & Jing-Mei Qiu

  2. Department of Mathematical Sciences, Rutgers University - Camden, Camden, NJ, 08102, USA

    Benedetto Piccoli

  3. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Tan Ren

Authors
  1. Suncica Canic
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  2. Benedetto Piccoli
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  3. Jing-Mei Qiu
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  4. Tan Ren
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Corresponding author

Correspondence to Tan Ren.

Additional information

The research of the first author is partially supported by NSF under Grants DMS-1263572, DMS-1318763, DMS-1311709, DMS-1262385, and DMS-1109189. The research of the second author is partially supported by NSF under Grant DMS-1107444. The research of the third and the fourth author is partially supported by Air Force Office of Scientific Computing YIP Grant FA9550-12-0318, NSF Grant DMS-1217008 and University of Houston.

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Canic, S., Piccoli, B., Qiu, JM. et al. Runge–Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks. J Sci Comput 63, 233–255 (2015). https://doi.org/10.1007/s10915-014-9896-z

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  • DOI: https://doi.org/10.1007/s10915-014-9896-z

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