American Institute of Mathematical Sciences

Advanced
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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (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.   Google Scholar

[14]

N. Kružhkov, First order quasilinear equations with several independent variables,, Matematicheskii Sbornik, 81 (1970), 228.   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.  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.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[17]

P. I. Richards, Shock waves on the highways,, Operational Research, 4 (1956), 42.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (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.   Google Scholar

[14]

N. Kružhkov, First order quasilinear equations with several independent variables,, Matematicheskii Sbornik, 81 (1970), 228.   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.  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.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[17]

P. I. Richards, Shock waves on the highways,, Operational Research, 4 (1956), 42.  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.  doi: 10.1006/jcph.1996.0053.  Google Scholar

[1]

Youshan Tao, J. Ignacio Tello. Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 193-207. doi: 10.3934/mbe.2016.13.193
[2]

Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains. Networks & Heterogeneous Media, 2014, 9 (3) : 501-518. doi: 10.3934/nhm.2014.9.501
[3]

Vanessa Baumgärtner, Simone Göttlich, Stephan Knapp. Feedback stabilization for a coupled PDE-ODE production system. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020003
[4]

Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028
[5]

Christophe Chalons, Paola Goatin, Nicolas Seguin. General constrained conservation laws. Application to pedestrian flow modeling. Networks & Heterogeneous Media, 2013, 8 (2) : 433-463. doi: 10.3934/nhm.2013.8.433
[6]

Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Numerical approximations of a traffic flow model on networks. Networks & Heterogeneous Media, 2006, 1 (1) : 57-84. doi: 10.3934/nhm.2006.1.57
[7]

Matthias Gerdts, Sven-Joachim Kimmerle. Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin. Conference Publications, 2015, 2015 (special) : 515-524. doi: 10.3934/proc.2015.0515
[8]

Mohamed Benyahia, Massimiliano D. Rosini. A macroscopic traffic model with phase transitions and local point constraints on the flow. Networks & Heterogeneous Media, 2017, 12 (2) : 297-317. doi: 10.3934/nhm.2017013
[9]

Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks & Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773
[10]

Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012
[11]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255
[12]

Adam Bobrowski, Katarzyna Morawska. From a PDE model to an ODE model of dynamics of synaptic depression. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2313-2327. doi: 10.3934/dcdsb.2012.17.2313
[13]

Debora Amadori, Wen Shen. Front tracking approximations for slow erosion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1481-1502. doi: 10.3934/dcds.2012.32.1481
[14]

Sabine Eisenhofer, Messoud A. Efendiev, Mitsuharu Ôtani, Sabine Schulz, Hans Zischka. On an ODE-PDE coupling model of the mitochondrial swelling process. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1031-1057. doi: 10.3934/dcdsb.2015.20.1031
[15]

Maria Laura Delle Monache, Paola Goatin. Stability estimates for scalar conservation laws with moving flux constraints. Networks & Heterogeneous Media, 2017, 12 (2) : 245-258. doi: 10.3934/nhm.2017010
[16]

Rinaldo M. Colombo, Andrea Corli. Dynamic parameters identification in traffic flow modeling. Conference Publications, 2005, 2005 (Special) : 190-199. doi: 10.3934/proc.2005.2005.190
[17]

Helge Holden, Nils Henrik Risebro. Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Networks & Heterogeneous Media, 2018, 13 (3) : 409-421. doi: 10.3934/nhm.2018018
[18]

Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control & Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121
[19]

Stefano Bianchini. On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 329-350. doi: 10.3934/dcds.2000.6.329
[20]

Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks & Heterogeneous Media, 2015, 10 (4) : 749-785. doi: 10.3934/nhm.2015.10.749

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences