# American Institute of Mathematical Sciences

June  2014, 7(3): 435-447. doi: 10.3934/dcdss.2014.7.435

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

 1 Inria Sophia Antipolis-Méditerranée - EPI OPALE, 2004, Route des Lucioles - BP 93, 06902 - Sophia Antipolis Cedex, France 2 INRIA Sophia Antipolis - Méditerranée, EPI OPALE, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex

Received  July 2013 Revised  August 2013 Published  January 2014

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.
Citation: Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435
##### References:
 [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. Goatin, Scalar Conservation Laws with Moving Density Constraints, INRIA Research Report, n.8119, 2012. Available from: , (). [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.

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##### References:
 [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. Goatin, Scalar Conservation Laws with Moving Density Constraints, INRIA Research Report, n.8119, 2012. Available from: , (). [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.
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