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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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

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