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Numerical discretization of Hamilton--Jacobi equations on networks

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  • We discuss a numerical discretization of Hamilton--Jacobi equations on networks. The latter arise for example as reformulation of the Lighthill--Whitham--Richards traffic flow model. We present coupling conditions for the Hamilton--Jacobi equations and derive a suitable numerical algorithm. Numerical computations of travel times in a round-about are given.
    Mathematics Subject Classification: Primary: 90B20, 35L50; Secondary: 35F21.

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  • [1]

    A. M. Bayen and C. G. Claudel, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations, SIAM J. Control Optim., 49 (2011), 383-402.doi: 10.1137/090778754.

    [2]

    A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM J. Math. Anal., 43 (2011), 2384-2417.doi: 10.1137/110825145.

    [3]

    G. Bretti, R. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on networks, Netw. Heterog. Media, 1 (2006), 57-84.doi: 10.3934/nhm.2006.1.57.

    [4]

    G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks, SIAM J. Appl. Dyn. Syst., 7 (2008), 510-531.doi: 10.1137/070697768.

    [5]

    Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 599-630.doi: 10.3934/dcdsb.2005.5.599.

    [6]

    G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.doi: 10.1137/S0036141004402683.

    [7]

    R. Corthout, G. Flötteröd, F. Viti and C. M. J. Tampère, Non-unique flows in macroscopic first-order intersection models, Transportation Res. Part B, 46 (2012), 343-359.doi: 10.1016/j.trb.2011.10.011.

    [8]

    C. F. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions, Transportation Res. Part B, 39 (2005), 187-196.doi: 10.1016/j.trb.2004.04.003.

    [9]

    C. F. Daganzo, The cell transmission model, part II: Network traffic, Transportation Res. Part B, 29 (1995), 79-93.doi: 10.1016/0191-2615(94)00022-R.

    [10]

    C. F. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications, Networks and Heterogeneous Media, 1 (2006), 601-619.doi: 10.3934/nhm.2006.1.601.

    [11]

    G. B. Dantzig, "Linear Programming and Extensions," Princeton University Press, Princeton, N.J. 1963 xvi+625 pp.

    [12]

    C. D'Apice, R. Manzo and B. Piccoli, A fluid dynamic model for telecommunication networks with sources and destinations, SIAM J. Appl. Math., 68 (2008), 981-1003.doi: 10.1137/060674132.

    [13]

    C. D'Apice, R. Manzo and L. Rarità, Splitting of traffic flows to control congestion in special events, Int. J. Math. Math. Sci., (2011), Art. ID 563171, 18 pages.doi: 10.1155/2011/563171.

    [14]

    G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transportation Res. Part B: Methodological, 45 (2011), 903-922 .

    [15]

    M. Garavello and B. Piccoli, "Traffic Flow on Networks," AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. xvi+243 pp.

    [16]

    M. Garavello and B. Piccoli, Source-destination flow on a road network, Commun. Math. Sci., 3 (2005), 261-283.

    [17]

    S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, (Russian) Mat. Sb. (N. S.), 47 (1959), 271-306.

    [18]

    B. Haut and G. Bastin, A second order model of road junctions in fluid models of traffic networks, Netw. Heterog. Media, 2 (2007), 227-253.doi: 10.3934/nhm.2007.2.227.

    [19]

    M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM Journal on Scientific Computing, 25 (2003), 1066-1087.doi: 10.1137/S106482750241459X.

    [20]

    M. Herty and M. Rascle, Coupling conditions for a class of "second-order'' models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616.doi: 10.1137/05062617X.

    [21]

    H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.doi: 10.1137/S0036141093243289.

    [22]

    C. Imbert, R. Monneau and H. Zidnani, A Hamilton-Jacobi approach to junction problems and application to traffic flow, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166.doi: 10.1051/cocv/2012002.

    [23]

    A. Kurganov and E. Tadmor, New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations, J. Comput. Phys., 160 (2000), 720-742.doi: 10.1006/jcph.2000.6485.

    [24]

    J.-P. Lebacque and M. Khoshyaran, First order macroscopic traffic flow models for networks in the context of dynamic assignment, Transportation Planning Applied Optimization, 64 (2004), 119-140.doi: 10.1007/0-306-48220-7_8.

    [25]

    R. J. LeVeque, "Numerical Methods for Conservation Laws," Second edition. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992.doi: 10.1007/978-3-0348-8629-1.

    [26]

    M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. Royal Society London. Ser. A., 229 (1955), 317-345.doi: 10.1098/rspa.1955.0089.

    [27]

    Y. Makigami, G. F. Newell and R. Rothery, Three-dimensional representation of traffic flow, Transportation Science, 5 (1971), 302-313.doi: 10.1287/trsc.5.3.302.

    [28]

    P. Mazarè, A. Dehwah, C. Claudel and A. Bayen, Analytical and grid-free solutions to the lighthill-whitham-richards traffic flow model, Transportation Res. Part B: Methodological, 45 (2011), 1727-1748.

    [29]

    K. Moskowitz, Discussion of freeway level of service as influenced by volume and capacity characteristics, Highway Research Record, 99 (1965), 43-44.

    [30]

    G. F. Newell, A simplified theory of kinematic waves in highway traffic: (I) general theory; (ii) queuing at freeway bottlenecks; (iii) multi-destination flow, Transportation Res. Part B, 27 (1993), 281-313.

    [31]

    P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.doi: 10.1287/opre.4.1.42.

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