# American Institute of Mathematical Sciences

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

 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

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.

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.

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##### 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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