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

TY - JOUR

T1 - Runge–Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

AU - Canic, Suncica

AU - Piccoli, Benedetto

AU - Qiu, Jing Mei

AU - Ren, Tan

N1 - Funding 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. Publisher Copyright: © 2014, Springer Science+Business Media New York.

PY - 2015/4

Y1 - 2015/4

N2 - 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.

AB - 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.

KW - Bound preserving

KW - Discontinuous Galerkin

KW - Hyperbolic network

KW - Scalar conservation laws

KW - Traffic flow

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U2 - https://doi.org/10.1007/s10915-014-9896-z

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

M3 - Article

SN - 0885-7474

VL - 63

SP - 233

EP - 255

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

IS - 1

ER -