A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. (English) Zbl 1292.90078

Summary: 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.


90B20 Traffic problems in operations research
65K05 Numerical mathematical programming methods
35L65 Hyperbolic conservation laws
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
Full Text: DOI


[1] B. Andreianov, Finite volume scheme for locally constrained conservation laws,, Numer. Math., 115, 609, (2010) · Zbl 1196.65151
[2] C. Bardos, First order quasilinear equations with boundary conditions,, Comm. Partial Differential Equations, 4, 1017, (1979) · Zbl 0418.35024
[3] R. Borsche, Mixed systems: ODEs - Balance laws,, Journal of Differential equations, 252, 2311, (2012) · Zbl 1252.35193
[4] B. Boutin, A convergent and conservative scheme for nonclassical solutions based on kinetic relations. I,, Interfaces and Free Boundaries, 10, 399, (2008) · Zbl 1157.65435
[5] G. Bretti, A tracking algorithm for car paths on road networks,, SIAM Journal of Applied Dynamical Systems, 7, 510, (2008) · Zbl 1168.35389
[6] C. Chalons, General constrained conservation laws. Application to pedestrian flow modeling,, Netw. Heterog. Media, 8, 433, (2013) · Zbl 1275.35144
[7] R. M. Colombo, A well posed conservation law with variable unilateral constraint,, Journal of Differential Equations, 234, 654, (2007) · Zbl 1253.65122
[8] R. M. Colombo, A Hölder continuous O.D.E. related to traffic flow,, The Royal Society of Edinburgh Proceedings A, 133, 759, (2003) · Zbl 1052.34007
[9] C. F. Daganzo, On the numerical treatement of moving bottlenecks,, Transportation Research Part B, 39, 31, (2005)
[10] C. F. Daganzo, Moving bottlenecks: A numerical method that converges in flows,, Transportation Research Part B, 39, 855, (2005)
[11] M. L. Delle Monache, <em>Scalar Conservation Laws with Moving Density Constraints</em>, INRIA Research Report, n.8119, 2012. Available from: <a href=
[12] Florence Giorgi, <em>Prise en Compte des Transports en Commune de Surface dans la Mod\'elisation Macroscopique de l’Écoulement du Trafic,</em>, Ph.D thesis, (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, 271, (1959) · Zbl 0171.46204
[14] N. Kružhkov, First order quasilinear equations with several independent variables,, Matematicheskii Sbornik, 81, 228, (1970) · Zbl 0202.11203
[15] C. Lattanzio, Moving bottlenecks in car traffic flow: a pde-ode coupled model,, SIAM Journal of Mathematical Analysis, 43, 50, (2011) · Zbl 1227.35206
[16] M. J. Lighthill, On kinetic waves. II. Theory of traffic flows on long crowded roads,, Proceedings of the Royal Society of London Series A, 229, 317, (1955) · Zbl 0064.20906
[17] P. I. Richards, Shock waves on the highways,, Operational Research, 4, 42, (1956) · Zbl 1414.90094
[18] X. Zhong, Computational Methods for propagating phase boundaries,, Journal of Computational Physics, 124, 192, (1996) · Zbl 0855.73080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.