\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow

Abstract Related Papers Cited by
  • In this paper we introduce a numerical method for tracking a bus trajectory on a road network. The mathematical model taken into consideration is a strongly coupled PDE-ODE system: the PDE is a scalar hyperbolic conservation law describing the traffic flow while the ODE, that describes the bus trajectory, needs to be intended in a Carathéodory sense. The moving constraint is given by an inequality on the flux which accounts for the bottleneck created by the bus on the road. The finite volume algorithm uses a locally non-uniform moving mesh which tracks the bus position. Some numerical tests are shown to describe the behavior of the solution.
    Mathematics Subject Classification: Primary: 58J45, 35L65; Secondary: 90B20.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    B. Andreianov, P. Goatin and N. Seguin, Finite volume scheme for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645.doi: 10.1007/s00211-009-0286-7.

    [2]

    C. Bardos, A. Y. LeRoux and J. C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034.doi: 10.1080/03605307908820117.

    [3]

    R. Borsche, R. M. Colombo and M. Garavello, Mixed systems: ODEs - Balance laws, Journal of Differential equations, 252 (2012), 2311-2338.doi: 10.1016/j.jde.2011.08.051.

    [4]

    B. Boutin, C. Chalons, F. Lagoutière and P. G. LeFloch, A convergent and conservative scheme for nonclassical solutions based on kinetic relations. I, Interfaces and Free Boundaries, 10 (2008), 399-421.doi: 10.4171/IFB/195.

    [5]

    G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks, SIAM Journal of Applied Dynamical Systems, 7 (2008), 510-531.doi: 10.1137/070697768.

    [6]

    C. Chalons, P. Goatin and N. Seguin, General constrained conservation laws. Application to pedestrian flow modeling, Netw. Heterog. Media, 8 (2013), 433-463.doi: 10.3934/nhm.2013.8.433.

    [7]

    R. M. Colombo and P. Goatin, A well posed conservation law with variable unilateral constraint, Journal of Differential Equations, 234 (2007), 654-675.doi: 10.1016/j.jde.2006.10.014.

    [8]

    R. M. Colombo and A. Marson, A Hölder continuous O.D.E. related to traffic flow, The Royal Society of Edinburgh Proceedings A, 133 (2003), 759-772.doi: 10.1017/S0308210500002663.

    [9]

    C. F. Daganzo and J. A. Laval, On the numerical treatement of moving bottlenecks, Transportation Research Part B, 39 (2005), 31-46.doi: 10.1016/j.trb.2004.02.003.

    [10]

    C. F. Daganzo and J. A. Laval, Moving bottlenecks: A numerical method that converges in flows, Transportation Research Part B, 39 (2005), 855-863.doi: 10.1016/j.trb.2004.10.004.

    [11]

    M. L. Delle Monache and P. GoatinScalar Conservation Laws with Moving Density Constraints, INRIA Research Report, n.8119 2012. Available from: http://hal.inria.fr/hal-00745671.

    [12]

    Florence Giorgi, Prise en Compte des Transports en Commune de Surface dans la Mod\'elisation Macroscopique de l'Écoulement du Trafic, Ph.D thesis, Insitut National des Sciences Appliquèes de Lyon, 2002.

    [13]

    S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Matematicheskii Sbornik, 47 (1959), 271-290.

    [14]

    N. Kružhkov, First order quasilinear equations with several independent variables, Matematicheskii Sbornik, 81 (1970), 228-255.

    [15]

    C. Lattanzio, A. Maurizi and B. Piccoli, Moving bottlenecks in car traffic flow: a pde-ode coupled model, SIAM Journal of Mathematical Analysis, 43 (2011), 50-67.doi: 10.1137/090767224.

    [16]

    M. J. Lighthill and G. B. Whitham, On kinetic waves. II. Theory of traffic flows on long crowded roads, Proceedings of the Royal Society of London Series A, 229 (1955), 317-345.doi: 10.1098/rspa.1955.0089.

    [17]

    P. I. Richards, Shock waves on the highways, Operational Research, 4 (1956), 42-51.doi: 10.1287/opre.4.1.42.

    [18]

    X. Zhong, T. Y. Hou and P. G. LeFloch, Computational Methods for propagating phase boundaries, Journal of Computational Physics, 124 (1996), 192-216.doi: 10.1006/jcph.1996.0053.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(177) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return