American Institute of Mathematical Sciences

Advanced
March  2017, 37(3): 1437-1487. doi: 10.3934/dcds.2017060

Homogenization of second order discrete model with local perturbation and application to traffic flow

Nicolas Forcadel , , Wilfredo Salazar and  Mamdouh Zaydan

Normandie Univ, INSA de Rouen Normandie, LMI (EA 3226 -FR CNRS 3335), 76000 Rouen, France, 685 Avenue de l'Université, 76801 St Etienne du Rouvray cedex, France

* Corresponding author: N. Forcadel

Received  May 2016 Revised  November 2016 Published  December 2016

The goal of this paper is to derive a traffic flow macroscopic model from a second order microscopic model with a local perturbation. At the microscopic scale, we consider a Bando model of the type following the leader, i.e the acceleration of each vehicle depends on the distance of the vehicle in front of it. We consider also a local perturbation like an accident at the roadside that slows down the vehicles. After rescaling, we prove that the "cumulative distribution functions" of the vehicles converges towards the solution of a macroscopic homogenized Hamilton-Jacobi equation with a flux limiting condition at junction which can be seen as a LWR (Lighthill-Whitham-Richards) model.

Keywords: Specified homogenization, Hamilton-Jacobi equations, traffic flow, integro-differential operators, Slepčcev formulation, viscosity solutions, microscopic models, macroscopic models.
Mathematics Subject Classification: 35D40, 90B20, 35B27, 35F20, 45K05.
Citation: Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Homogenization of second order discrete model with local perturbation and application to traffic flow. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1437-1487. doi: 10.3934/dcds.2017060
References:
[1]

Y. Achdou and N. Tchou, Hamilton-jacobi equations on networks as limits of singularly perturbed problems in optimal control: Dimension reduction, Communications in Partial Differential Equations, 40 (2015), 652-693.  doi: 10.1080/03605302.2014.974764.  Google Scholar

[2]

O. Alvarez and A. Tourin, Viscosity solutions of nonlinear integro-differential equations, Annales de l'Institut Henri Poincaré. Analyse non linéaire, 13 (1996), 293-317.  doi: 10.1016/j.anihpc.2007.02.007.  Google Scholar

[3]

A. Aw, A. Klar, M. Rascle and T. Materne, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM Journal on Applied Mathematics, 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.  Google Scholar

[4]

M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation Physical Review E 51 (1995), p1035. doi: 10.1103/PhysRevE.51.1035.  Google Scholar

[5]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi Springer Verlag, 1994.  Google Scholar

[6]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[7]

F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061-1104.  doi: 10.4171/JEMS/140.  Google Scholar

[8]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4.  Google Scholar

[9]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of fully overdamped frenkel-kontorova models, Journal of Differential Equations, 246 (2009), 1057-1097.  doi: 10.1016/j.jde.2008.06.034.  Google Scholar

[10]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, Discrete Contin. Dyn. Syst., 23 (2009), 785-826.  doi: 10.3934/dcds.2009.23.785.  Google Scholar

[11]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of accelerated frenkel-kontorova models with n types of particles, Transactions of the American Mathematical Society, 364 (2012), 6187-6227.  doi: 10.1090/S0002-9947-2012-05650-9.  Google Scholar

[12]

N. Forcadel and W. Salazar, Homogenization of second order discrete model and application to traffic flow, Differential and Integral Equations, 28 (2015), 1039-1068.   Google Scholar

[13]

N. Forcadel and W. Salazar, A junction condition by specified homogenization of a discrete model with a local perturbation and application to traffic flow, preprint, hal-01097085. Google Scholar

[14]

G. Galise, C. Imbert and R. Monneau, A junction condition by specified homogenization and application to traffic lights, Anal. PDE, 8 (2015), 1891-1929.  doi: 10.2140/apde.2015.8.1891.  Google Scholar

[15]

J. M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascale, SIAM J. Appl. Math., 62 (2001), 729-745.  doi: 10.1137/S0036139900378657.  Google Scholar

[16]

D. Helbing, From microscopic to macroscopic traffic models, in A Perspective Look at Nonlinear Media, Lecture Notes in Phys. , 503, Springer, Berlin, 1998,122–139. doi: 10.1007/BFb0104959.  Google Scholar

[17]

C. Imbert, A non-local regularization of first order Hamilton--Jacobi equations, Journal of Differential Equations, 211 (2005), 218-246.  doi: 10.1016/j.jde.2004.06.001.  Google Scholar

[18]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179.  doi: 10.3934/krm.2010.3.165.  Google Scholar

[19]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex hamilton-jacobi equations on networks, arXiv: 1306.2428. Google Scholar

[20]

C. Imbert, R. Monneau and E. Rouy, Homogenization of first order equations with (u/$\varepsilon$)-periodic hamiltonians part ⅱ: Application to dislocations dynamics, Communications in Partial Differential Equations, 33 (2008), 479-516.  doi: 10.1080/03605300701318922.  Google Scholar

[21]

H. Ishii and S. Koike, Viscosity solutions for monotone systems of second--order elliptic pdes, Communications in Partial Differential Equations, 16 (1991), 1095-1128.  doi: 10.1080/03605309108820791.  Google Scholar

[22]

W. Knödel, Graphentheoretische {M}ethoden und Ihre {A}nwendungen Econometrics and Operations Research, ⅩⅢ, Springer-Verlag, Berlin-New York, 1969. doi: 10.1007/978-3-642-95121-3.  Google Scholar

[23]

H. Lee, H. -W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models Physical Review E 64 (2001), 056126. doi: 10.1103/PhysRevE.64.056126.  Google Scholar

[24]

M. J. Lighthill and G. B. Whitham, On kinematic waves. ⅱ. a theory of traffic flow on long crowded roadss, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1995), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[25]

P. L. Lions, Lectures at collège de france, 2013-2014. Google Scholar

[26]

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

show all references

References:
[1]

Y. Achdou and N. Tchou, Hamilton-jacobi equations on networks as limits of singularly perturbed problems in optimal control: Dimension reduction, Communications in Partial Differential Equations, 40 (2015), 652-693.  doi: 10.1080/03605302.2014.974764.  Google Scholar

[2]

O. Alvarez and A. Tourin, Viscosity solutions of nonlinear integro-differential equations, Annales de l'Institut Henri Poincaré. Analyse non linéaire, 13 (1996), 293-317.  doi: 10.1016/j.anihpc.2007.02.007.  Google Scholar

[3]

A. Aw, A. Klar, M. Rascle and T. Materne, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM Journal on Applied Mathematics, 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.  Google Scholar

[4]

M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation Physical Review E 51 (1995), p1035. doi: 10.1103/PhysRevE.51.1035.  Google Scholar

[5]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi Springer Verlag, 1994.  Google Scholar

[6]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[7]

F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061-1104.  doi: 10.4171/JEMS/140.  Google Scholar

[8]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4.  Google Scholar

[9]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of fully overdamped frenkel-kontorova models, Journal of Differential Equations, 246 (2009), 1057-1097.  doi: 10.1016/j.jde.2008.06.034.  Google Scholar

[10]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, Discrete Contin. Dyn. Syst., 23 (2009), 785-826.  doi: 10.3934/dcds.2009.23.785.  Google Scholar

[11]

N. Forcadel, C. Imbert and R. Monneau, Homogenization of accelerated frenkel-kontorova models with n types of particles, Transactions of the American Mathematical Society, 364 (2012), 6187-6227.  doi: 10.1090/S0002-9947-2012-05650-9.  Google Scholar

[12]

N. Forcadel and W. Salazar, Homogenization of second order discrete model and application to traffic flow, Differential and Integral Equations, 28 (2015), 1039-1068.   Google Scholar

[13]

N. Forcadel and W. Salazar, A junction condition by specified homogenization of a discrete model with a local perturbation and application to traffic flow, preprint, hal-01097085. Google Scholar

[14]

G. Galise, C. Imbert and R. Monneau, A junction condition by specified homogenization and application to traffic lights, Anal. PDE, 8 (2015), 1891-1929.  doi: 10.2140/apde.2015.8.1891.  Google Scholar

[15]

J. M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascale, SIAM J. Appl. Math., 62 (2001), 729-745.  doi: 10.1137/S0036139900378657.  Google Scholar

[16]

D. Helbing, From microscopic to macroscopic traffic models, in A Perspective Look at Nonlinear Media, Lecture Notes in Phys. , 503, Springer, Berlin, 1998,122–139. doi: 10.1007/BFb0104959.  Google Scholar

[17]

C. Imbert, A non-local regularization of first order Hamilton--Jacobi equations, Journal of Differential Equations, 211 (2005), 218-246.  doi: 10.1016/j.jde.2004.06.001.  Google Scholar

[18]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179.  doi: 10.3934/krm.2010.3.165.  Google Scholar

[19]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex hamilton-jacobi equations on networks, arXiv: 1306.2428. Google Scholar

[20]

C. Imbert, R. Monneau and E. Rouy, Homogenization of first order equations with (u/$\varepsilon$)-periodic hamiltonians part ⅱ: Application to dislocations dynamics, Communications in Partial Differential Equations, 33 (2008), 479-516.  doi: 10.1080/03605300701318922.  Google Scholar

[21]

H. Ishii and S. Koike, Viscosity solutions for monotone systems of second--order elliptic pdes, Communications in Partial Differential Equations, 16 (1991), 1095-1128.  doi: 10.1080/03605309108820791.  Google Scholar

[22]

W. Knödel, Graphentheoretische {M}ethoden und Ihre {A}nwendungen Econometrics and Operations Research, ⅩⅢ, Springer-Verlag, Berlin-New York, 1969. doi: 10.1007/978-3-642-95121-3.  Google Scholar

[23]

H. Lee, H. -W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models Physical Review E 64 (2001), 056126. doi: 10.1103/PhysRevE.64.056126.  Google Scholar

[24]

M. J. Lighthill and G. B. Whitham, On kinematic waves. ⅱ. a theory of traffic flow on long crowded roadss, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1995), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[25]

P. L. Lions, Lectures at collège de france, 2013-2014. Google Scholar

[26]

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

[1]

Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389
[2]

Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure & Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793
[3]

Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649
[4]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461
[5]

Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677
[6]

Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176
[7]

Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064
[8]

Gabriella Puppo, Matteo Semplice, Andrea Tosin, Giuseppe Visconti. Kinetic models for traffic flow resulting in a reduced space of microscopic velocities. Kinetic & Related Models, 2017, 10 (3) : 823-854. doi: 10.3934/krm.2017033
[9]

Michael Herty, Reinhard Illner. Analytical and numerical investigations of refined macroscopic traffic flow models. Kinetic & Related Models, 2010, 3 (2) : 311-333. doi: 10.3934/krm.2010.3.311
[10]

Paola Goatin, Elena Rossi. Comparative study of macroscopic traffic flow models at road junctions. Networks & Heterogeneous Media, 2020, 15 (2) : 261-279. doi: 10.3934/nhm.2020012
[11]

Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385
[12]

Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647
[13]

Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167
[14]

David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205
[15]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
[16]

Xu Chen, Jianping Wan. Integro-differential equations for foreign currency option prices in exponential Lévy models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 529-537. doi: 10.3934/dcdsb.2007.8.529
[17]

Eddaly Guerra, Héctor Sánchez-Morgado. Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 331-346. doi: 10.3934/cpaa.2014.13.331
[18]

Guillaume Costeseque, Jean-Patrick Lebacque. Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 411-433. doi: 10.3934/dcdss.2014.7.411
[19]

Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513
[20]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (209)
  • HTML views (50)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy