# American Institute of Mathematical Sciences

September  2018, 17(5): 2173-2206. doi: 10.3934/cpaa.2018104

## Specified homogenization of a discrete traffic model leading to an effective junction condition

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

* Corresponding author

Received  June 2017 Revised  January 2018 Published  April 2018

In this paper, we focus on deriving traffic flow macroscopic models from microscopic models containing a local perturbation such as a traffic light. At the microscopic scale, we consider a first order model of the form "follow the leader" i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider a local perturbation located at the origin that slows down the vehicles. At the macroscopic scale, we obtain an explicit Hamilton-Jacobi equation left and right of the origin and a junction condition at the origin (in the sense of [25]) which keeps the memory of the local perturbation. As it turns out, the macroscopic model is equivalent to a LWR model, with a flux limiting condition at the junction. Finally, we also present qualitative properties concerning the flux limiter at the junction.

Citation: Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Specified homogenization of a discrete traffic model leading to an effective junction condition. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2173-2206. doi: 10.3934/cpaa.2018104
##### 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. [2] Y. Achdou, F. Camilli, A. Cutrì and N. Tchou, Hamilton-Jacobi equations constrained on networks, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 413-445. [3] Y. Achdou, S. Oudet and N. Tchou, Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1091-1121. [4] 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. [5] 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. [6] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $R^N$, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 710-739. [7] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $R^N$, SIAM Journal on Control and Optimization, 52 (2014), 1712-1744. [8] G. Barles, A. Briani, E. Chasseigne and C. Imbert, Flux-limited and classical viscosity solutions for regional control problems, preprint, arXiv: math/1611.01977. [9] M. Batista and E. Twrdy, Optimal velocity functions for car-following models, Journal of Zhejiang University-SCIENCE A, 11 (2010), 520-529. [10] F. Camilli, C. Marchi and D. Schieborn, The vanishing viscosity limit for Hamilton-Jacobi equations on networks, Journal of Differential Equations, 254 (2013), 4122-4143. [11] E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, preprint, arXiv: math/1505.01372, [12] M. Di Francesco and M.D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for rational mechanics and analysis, 217 (2015), 831-871. [13] C. Edie, Car-following and steady-state theory for noncongested traffic, Operations Research, 9 (1961), 66-76. [14] L. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 111 (1989), 359-375. [15] N. Forcadel, C. Imbert and R. Monneau, Homogenization of fully overdamped Frenkel-Kontorova models, Journal of Differential Equations, 246 (2009), 1057-1097. [16] N. Forcadel and W. Salazar, Homogenization of second order discrete model and application to traffic flow, Differential and Integral Equations, 28 (2015), 1039-1068. [17] 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. [18] N. Forcadel, C. Imbert and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, Discrete and Continuous Dynamical Systems-Series A, 23 (2009), 785-826. [19] N. Forcadel, W. Salazar and M. Zaydan, Homogenization of second order discrete model with local perturbation and application to traffic flow, Discrete and Continuous Dynamical Systems-Series A, 37 (2017), 1437-1487. [20] G. Galise, C. Imbert and R. Monneau, A junction condition by specified homogenization and application to traffic lights, Analysis & PDE, 8 (2015), 1891-1929. [21] BD Greenshields, Ws Channing, Hh. Miller and others, A study of traffic capacity, in Highway research board proceedings, (1935). [22] D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro-and macro-simulation of freeway traffic, Mathematical and Computer Modelling, 35 (2002), 517-547. [23] C. Imbert, A non-local regularization of first order Hamilton-Jacobi equations, Journal of Differential Equations, 211 (2005), 218-246. [24] C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case, Discrete and Continuous Dynamical Systems-Series A, 37 (2014), 6405-6435. [25] C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Annales Scientifiques de l'ENS, 50 (2017), 357-448. [26] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 129-166. [27] 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. [28] J. P. Lebacque and M. M. Khoshyaran, Modelling vehicular traffic flow on networks using macroscopic models, Finite Volumes for Complex Applications Ⅱ, (1999), 551-558. [29] H. K. Lee, H. W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models, Physical Review E, 64 (2001), 056126. [30] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 229 (1955), 317-345. [31] P. L. Lions, Lectures at Collège de France, (2015-2016). [32] P. L. Lions, Lectures at Collège de France, (2013-2014). [33] P. L. Lions and P. E. Souganidis, Viscosity solutions for junctions: well posedness and stability, Rendiconti Lincei-Matematica e Applicazioni, 27 (2016), 535-545. [34] G. F. Newell, Nonlinear effects in the dynamics of car following, Operations research, 9 (1961), 209-229. [35] H. J. Payne, Models of freeway traffic and control, Mathematical Models of Public Systems, (1971). [36] P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. [37] E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete & Continuous Dynamical Systems-Series S, 7 (2014). [38] W. Salazar, Numerical specified homogenization of a discrete model with a local perturbation and application to traffic flow, (2016). [39] D. Slepčev, Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 52 (2003), 79-115.

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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. [2] Y. Achdou, F. Camilli, A. Cutrì and N. Tchou, Hamilton-Jacobi equations constrained on networks, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 413-445. [3] Y. Achdou, S. Oudet and N. Tchou, Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1091-1121. [4] 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. [5] 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. [6] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $R^N$, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 710-739. [7] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $R^N$, SIAM Journal on Control and Optimization, 52 (2014), 1712-1744. [8] G. Barles, A. Briani, E. Chasseigne and C. Imbert, Flux-limited and classical viscosity solutions for regional control problems, preprint, arXiv: math/1611.01977. [9] M. Batista and E. Twrdy, Optimal velocity functions for car-following models, Journal of Zhejiang University-SCIENCE A, 11 (2010), 520-529. [10] F. Camilli, C. Marchi and D. Schieborn, The vanishing viscosity limit for Hamilton-Jacobi equations on networks, Journal of Differential Equations, 254 (2013), 4122-4143. [11] E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, preprint, arXiv: math/1505.01372, [12] M. Di Francesco and M.D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for rational mechanics and analysis, 217 (2015), 831-871. [13] C. Edie, Car-following and steady-state theory for noncongested traffic, Operations Research, 9 (1961), 66-76. [14] L. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 111 (1989), 359-375. [15] N. Forcadel, C. Imbert and R. Monneau, Homogenization of fully overdamped Frenkel-Kontorova models, Journal of Differential Equations, 246 (2009), 1057-1097. [16] N. Forcadel and W. Salazar, Homogenization of second order discrete model and application to traffic flow, Differential and Integral Equations, 28 (2015), 1039-1068. [17] 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. [18] N. Forcadel, C. Imbert and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, Discrete and Continuous Dynamical Systems-Series A, 23 (2009), 785-826. [19] N. Forcadel, W. Salazar and M. Zaydan, Homogenization of second order discrete model with local perturbation and application to traffic flow, Discrete and Continuous Dynamical Systems-Series A, 37 (2017), 1437-1487. [20] G. Galise, C. Imbert and R. Monneau, A junction condition by specified homogenization and application to traffic lights, Analysis & PDE, 8 (2015), 1891-1929. [21] BD Greenshields, Ws Channing, Hh. Miller and others, A study of traffic capacity, in Highway research board proceedings, (1935). [22] D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro-and macro-simulation of freeway traffic, Mathematical and Computer Modelling, 35 (2002), 517-547. [23] C. Imbert, A non-local regularization of first order Hamilton-Jacobi equations, Journal of Differential Equations, 211 (2005), 218-246. [24] C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case, Discrete and Continuous Dynamical Systems-Series A, 37 (2014), 6405-6435. [25] C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Annales Scientifiques de l'ENS, 50 (2017), 357-448. [26] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 129-166. [27] 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. [28] J. P. Lebacque and M. M. Khoshyaran, Modelling vehicular traffic flow on networks using macroscopic models, Finite Volumes for Complex Applications Ⅱ, (1999), 551-558. [29] H. K. Lee, H. W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models, Physical Review E, 64 (2001), 056126. [30] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 229 (1955), 317-345. [31] P. L. Lions, Lectures at Collège de France, (2015-2016). [32] P. L. Lions, Lectures at Collège de France, (2013-2014). [33] P. L. Lions and P. E. Souganidis, Viscosity solutions for junctions: well posedness and stability, Rendiconti Lincei-Matematica e Applicazioni, 27 (2016), 535-545. [34] G. F. Newell, Nonlinear effects in the dynamics of car following, Operations research, 9 (1961), 209-229. [35] H. J. Payne, Models of freeway traffic and control, Mathematical Models of Public Systems, (1971). [36] P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. [37] E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete & Continuous Dynamical Systems-Series S, 7 (2014). [38] W. Salazar, Numerical specified homogenization of a discrete model with a local perturbation and application to traffic flow, (2016). [39] D. Slepčev, Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 52 (2003), 79-115.
Schematic representation of the microscopic model
Schematic representation of the macroscopic model
Schematic representation of the optimal velocity function $V$
Schematic representation of $\bar{H}$
Schematic representation of the function $\rho$
Schematic representation of the function $\rho$
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