Representation of capacity drop at a road merge via point constraints in a first order traffic model
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 1-34.

We reproduce the capacity drop phenomenon at a road merge by implementing a non-local point constraint at the junction in a first order traffic model. We call capacity drop the situation in which the outflow through the junction is lower than the receiving capacity of the outgoing road, as too many vehicles trying to access the junction from the incoming roads hinder each other. In this paper, we first construct an enhanced version of the locally constrained model introduced by Haut et al. (Proceedings 16th IFAC World Congress. Prague, Czech Republic 229 (2005) TuM01TP/3), then we propose its counterpart featuring a non-local constraint and finally we compare numerically the two models by constructing an adapted finite volumes scheme.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2019002
Classification : 35L65, 35R02, 90B20, 76M12
Mots-clés : Scalar conservation law, LWR model, traffic flow on networks, point constraint on the flux, finite volumes schemes
Dal Santo, Edda 1 ; Donadello, Carlotta 1 ; Pellegrino, Sabrina F. 1 ; Rosini, Massimiliano D. 1

1 
@article{M2AN_2019__53_1_1_0,
     author = {Dal Santo, Edda and Donadello, Carlotta and Pellegrino, Sabrina F. and Rosini, Massimiliano D.},
     title = {Representation of capacity drop at a road merge via point constraints in a first order traffic model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1--34},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {1},
     year = {2019},
     doi = {10.1051/m2an/2019002},
     zbl = {1420.35156},
     mrnumber = {3922818},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2019002/}
}
TY  - JOUR
AU  - Dal Santo, Edda
AU  - Donadello, Carlotta
AU  - Pellegrino, Sabrina F.
AU  - Rosini, Massimiliano D.
TI  - Representation of capacity drop at a road merge via point constraints in a first order traffic model
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2019
SP  - 1
EP  - 34
VL  - 53
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2019002/
DO  - 10.1051/m2an/2019002
LA  - en
ID  - M2AN_2019__53_1_1_0
ER  - 
%0 Journal Article
%A Dal Santo, Edda
%A Donadello, Carlotta
%A Pellegrino, Sabrina F.
%A Rosini, Massimiliano D.
%T Representation of capacity drop at a road merge via point constraints in a first order traffic model
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2019
%P 1-34
%V 53
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2019002/
%R 10.1051/m2an/2019002
%G en
%F M2AN_2019__53_1_1_0
Dal Santo, Edda; Donadello, Carlotta; Pellegrino, Sabrina F.; Rosini, Massimiliano D. Representation of capacity drop at a road merge via point constraints in a first order traffic model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 1-34. doi : 10.1051/m2an/2019002. http://www.numdam.org/articles/10.1051/m2an/2019002/

[1] B. Andreianov, G.M. Coclite and C. Donadello, Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete Contin. Dyn. Syst. A 37 (2017) 5913–5942. | DOI | MR | Zbl

[2] B. Andreianov, C. Donadello, U. Razafison and M.D. Rosini, Riemann problems with non–local point constraints and capacity drop. Math. Biosci. Eng. 12 (2015) 259–278. | DOI | MR | Zbl

[3] B. Andreianov, C. Donadello, U. Razafison and M.D. Rosini, Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks. ESAIM: M2AN 50 (2016) 1269–1287. | DOI | Numdam | MR | Zbl

[4] B. Andreianov, C. Donadello, U. Razafison and M.D. Rosini, Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux. J. Math. Pures App. 116 (2018) 309–346. | DOI | MR | Zbl

[5] B. Andreianov, C. Donadello and M.D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop. Math. Models Methods Appl. Sci. 24 (2014) 2685–2722. | DOI | MR | Zbl

[6] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws. Numer. Math. 115 (2010) 609–645. | DOI | MR | Zbl

[7] C. Bardos, A.Y. Leroux and J.C. Nedelec, First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4 (1979) 1017–1034. | DOI | MR | Zbl

[8] A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli, Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1 (2014) 47–111. | DOI | MR | Zbl

[9] C. Cancès and N. Seguin, Error estimate for Godunov approximation of locally constrained conservation laws. SIAM J. Numer. Anal. 50 (2012) 3036–3060. | DOI | MR | Zbl

[10] C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows. SIAM J. Sci. Comput. 29 (2007) 539–555. | DOI | MR | Zbl

[11] C. Chalons, P. Goatin and N. Seguin, General constrained conservation laws. Application to pedestrian flow modeling.. Networks Heterogen. Media 8 (2013) 433–463. | DOI | MR | Zbl

[12] G.M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network. SIAM J. Math. Anal. 36 (2005) 1862–1886. | DOI | MR | Zbl

[13] R. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234 (2007) 654–675. | DOI | MR | Zbl

[14] R.M. Colombo, P. Goatin and M.D. Rosini, A macroscopic model for pedestrian flows in panic situations. In: Current Advances in Nonlinear Analysis and Related Topics. GAKUTO Internat. Ser. Math. Sci. Appl. Gakkōtosho, Tokyo 32 (2010) 255–272. | MR | Zbl

[15] R.M. Colombo, P. Goatin and M.D. Rosini, On the modelling and management of traffic. ESAIM: M2AN 45 (2011) 853–872. | DOI | Numdam | MR | Zbl

[16] R.M. Colombo and M.D. Rosini, Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci. 28 (2005) 1553–1567. | DOI | MR | Zbl

[17] R.M. Colombo and M.D. Rosini, Existence of nonclassical solutions in a pedestrian flow model. Nonlinear Anal.: Real World App. 10 (2009) 2716–2728. | DOI | MR | Zbl

[18] C. D’Apice, R. Manzo and B. Piccoli, Packet flow on telecommunication networks. SIAM J. Math. Anal. 38 (2006) 717–740. | DOI | MR | Zbl

[19] M. Garavello, K. Han and B. Piccoli, Models for vehicular traffic on networks. Conservation laws models. AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO 9 (2016). | MR | Zbl

[20] M. Garavello and B. Piccoli, Traffic flow on networks. Conservation laws models. AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO 1 (2006). | MR | Zbl

[21] M. Garavello and B. Piccoli, Conservation laws on complex networks. Ann. Inst. Henri Poincaré (C) Non Lin. Anal. 26 (2009) 1925–1951. | DOI | Numdam | MR | Zbl

[22] B. Haut, G. Bastin and Y. Chitour, A macroscopic traffic model for road networks with a representation of the capacity drop phenomenon at the junctions. In Vol. 226 of Proceedings 16th IFAC World Congress, Prague, Czech Republic (2005) Tu–M01–TP/3.

[23] S.N. Kruzhkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228–255. | MR | Zbl

[24] M.J. Lighthill and G.B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A. 229 (1955) 317–345. | DOI | MR | Zbl

[25] E.Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4 (2007) 729–770. | DOI | MR | Zbl

[26] S.F. Pellegrino, On the implementation of a finite volumes scheme with monotone transmission conditions for scalar conservation laws on a star-shaped network. Preprint arXiv:1902.02395 (2019). | MR

[27] P.I. Richards, Shock waves on the highway. Oper. Res. 4 (1956) 42–51. | DOI | MR | Zbl

[28] M.D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications. Springer, Heidelberg (2013). | DOI | MR | Zbl

[29] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal. 160 (2001) 181–193. | DOI | MR | Zbl

Cité par Sources :