American Institute of Mathematical Sciences

Advanced
June  2014, 7(3): 579-591. doi: 10.3934/dcdss.2014.7.579

A justification of a LWR model based on a follow the leader description

Elena Rossi 1,

1. 

Department of Mathematics and Applications, University of Milano-Bicocca, Via Cozzi, 53, 20125 Milano, Italy

Received  May 2013 Revised  July 2013 Published  January 2014

We investigate the correlations between a macroscopic Lighthill--Whitham and Richards model and a microscopic follow-the-leader model for traffic flow. We prove that the microscopic model tends to the macroscopic one in a sort of kinetic limit, i.e. as the number of individuals tends to infinity, keeping the total mass fixed. Based on this convergence result, we approximately compute the solutions to a conservation law by means of the integration of an ordinary differential system. From the numerical point of view, the limiting procedure is then extended to the case of several populations, referring to the macroscopic model in [2] and to the natural multi--population analogue of the microscopic one.
Keywords: Macroscopic traffic models, microscopic to macoscopic limit., microscopic traffic models, conservation laws, follow the leader model.
Mathematics Subject Classification: Primary: 35L65; Secondary: 90B2.
Citation: Elena Rossi. A justification of a LWR model based on a follow the leader description. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 579-591. doi: 10.3934/dcdss.2014.7.579
References:
[1]

B. Argall, E. Cheleshkin, J. M. Greenberg, C. Hinde and P.-J. Lin, A rigorous treatment of a follow-the-leader traffic model with traffic lights present,, SIAM J. Appl. Math., 63 (2002), 149.  doi: 10.1137/S0036139901391215.  Google Scholar

[2]

S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow,, European J. Appl. Math., 14 (2003), 587.  doi: 10.1017/S0956792503005266.  Google Scholar

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws,, Oxford Lecture Series in Mathematics and its Applications, (2000).   Google Scholar

[4]

R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow,, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759.  doi: 10.1017/S0308210500002663.  Google Scholar

[5]

R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound,, SIAM J. Appl. Math., 70 (2010), 2652.  doi: 10.1137/090752468.  Google Scholar

[6]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow,, to appear in Rend. Semin. Mat. Univ. Padova., ().   Google Scholar

[7]

G. Costeseque, Analyse et Modelisation du Trafic Routier: Passage du Microscopique au Macroscopique,, Master's Thesis, (2011).   Google Scholar

[8]

S. N. Kružkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (1970), 228.   Google Scholar

[9]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge Texts in Applied Mathematics, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[10]

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

[11]

K. Radhakrishnan and A. C. Hindmarsh, Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations,, LLNL report UCRL-ID-113855, (1993).  doi: 10.2172/15013302.  Google Scholar

[12]

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

[13]

E. Rossi, On the Micro-Macro Limit in Traffic Flow,, Master's thesis, (2012).   Google Scholar

show all references

References:
[1]

B. Argall, E. Cheleshkin, J. M. Greenberg, C. Hinde and P.-J. Lin, A rigorous treatment of a follow-the-leader traffic model with traffic lights present,, SIAM J. Appl. Math., 63 (2002), 149.  doi: 10.1137/S0036139901391215.  Google Scholar

[2]

S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow,, European J. Appl. Math., 14 (2003), 587.  doi: 10.1017/S0956792503005266.  Google Scholar

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws,, Oxford Lecture Series in Mathematics and its Applications, (2000).   Google Scholar

[4]

R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow,, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759.  doi: 10.1017/S0308210500002663.  Google Scholar

[5]

R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound,, SIAM J. Appl. Math., 70 (2010), 2652.  doi: 10.1137/090752468.  Google Scholar

[6]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow,, to appear in Rend. Semin. Mat. Univ. Padova., ().   Google Scholar

[7]

G. Costeseque, Analyse et Modelisation du Trafic Routier: Passage du Microscopique au Macroscopique,, Master's Thesis, (2011).   Google Scholar

[8]

S. N. Kružkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (1970), 228.   Google Scholar

[9]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge Texts in Applied Mathematics, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[10]

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

[11]

K. Radhakrishnan and A. C. Hindmarsh, Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations,, LLNL report UCRL-ID-113855, (1993).  doi: 10.2172/15013302.  Google Scholar

[12]

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

[13]

E. Rossi, On the Micro-Macro Limit in Traffic Flow,, Master's thesis, (2012).   Google Scholar

[1]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[2]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[3]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170
[4]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345
[5]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010
[6]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[7]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304
[8]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168
[9]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460
[10]

Yen-Luan Chen, Chin-Chih Chang, Zhe George Zhang, Xiaofeng Chen. Optimal preventive "maintenance-first or -last" policies with generalized imperfect maintenance models. Journal of Industrial & Management Optimization, 2021, 17 (1) : 501-516. doi: 10.3934/jimo.2020149
[11]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[12]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457
[13]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464
[14]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[15]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[16]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317
[17]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219
[18]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344
[19]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349
[20]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences