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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

Maria Laura Delle Monache 1, and  Paola Goatin 2,

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.
Keywords: PDE-ODE model, Conservation laws with constraints, numerical simulations, front-tracking methods., traffic flow modeling.
Mathematics Subject Classification: Primary: 58J45, 35L65; Secondary: 90B2.
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 & 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.  Google Scholar

[2]

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. L. Delle Monache and P. Goatin, Scalar Conservation Laws with Moving Density Constraints, INRIA Research Report, n.8119, 2012. Available from: , ().   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. L. Delle Monache and P. Goatin, Scalar Conservation Laws with Moving Density Constraints, INRIA Research Report, n.8119, 2012. Available from: , ().   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

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