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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 |
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.
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. 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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. Google Scholar |
[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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