We present a network formulation for a traffic flow model with nonlocal velocity in the flux function. The modeling framework includes suitable coupling conditions at intersections to either ensure maximum flux or distribution parameters. In particular, we focus on 1-to-1, 2-to-1 and 1-to-2 junctions. Based on an upwind type numerical scheme, we prove the maximum principle and the existence of weak solutions on networks. We also investigate the limiting behavior of the proposed models when the nonlocal influence tends to infinity. Numerical examples show the difference between the proposed coupling conditions and a comparison to the Lighthill-Whitham-Richards network model.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2022002
Keywords: Nonlocal scalar conservation laws, traffic flow networks, coupling conditions, upwind scheme
@article{M2AN_2022__56_1_213_0,
author = {Friedrich, Jan and G\"ottlich, Simone and Osztfalk, Maximilian},
title = {Network models for nonlocal traffic flow},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {213--235},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {1},
doi = {10.1051/m2an/2022002},
mrnumber = {4376274},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022002/}
}
TY - JOUR AU - Friedrich, Jan AU - Göttlich, Simone AU - Osztfalk, Maximilian TI - Network models for nonlocal traffic flow JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 213 EP - 235 VL - 56 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022002/ DO - 10.1051/m2an/2022002 LA - en ID - M2AN_2022__56_1_213_0 ER -
%0 Journal Article %A Friedrich, Jan %A Göttlich, Simone %A Osztfalk, Maximilian %T Network models for nonlocal traffic flow %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 213-235 %V 56 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022002/ %R 10.1051/m2an/2022002 %G en %F M2AN_2022__56_1_213_0
Friedrich, Jan; Göttlich, Simone; Osztfalk, Maximilian. Network models for nonlocal traffic flow. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 1, pp. 213-235. doi: 10.1051/m2an/2022002
[1] , and , A model for the dynamics of large queuing networks and supply chains. SIAM J. Appl. Math. 66 (2006) 896–920. | MR | Zbl | DOI
[2] and , Resurrection of ``second order’’ models of traffic flow. SIAM J. Appl. Math. 60 (2000) 916–938. | MR | Zbl | DOI
[3] , , and , Modeling multi-lane traffic with moving obstacles by nonlocal balance laws. Preprint (2020). | MR
[4] and , Well-posedness of a conservation law with non-local flux arising in traffic flow modeling. Numer. Math. 132 (2016) 217–241. | MR | DOI
[5] and , Entropy admissibility of the limit solution for a nonlocal model of traffic flow. Preprint (2020). | arXiv | MR
[6] and , On traffic flow with nonlocal flux: a relaxation representation. Arch. Ration. Mech. Anal. 237 (2020) 1213–1236. | MR | DOI
[7] , and , Measure-valued solutions to nonlocal transport equations on networks. J. Differ. Equ. 264 (2018) 7213–7241. | MR | DOI
[8] , and , High-order numerical schemes for one-dimensional nonlocal conservation laws. SIAM J. Sci. Comput. 40 (2018) A288–A305. | MR | DOI
[9] and , Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel. ESAIM: M2AN 52 (2018) 163–180. | MR | Zbl | Numdam | DOI
[10] and , Non-local multi-class traffic flow models. Netw. Heterog. Media 14 (2019) 371–387. | MR | DOI
[11] , , , and , A non-local traffic flow model for 1-to-1 junctions. Eur. J. Appl. Math. (2019) 1–21. | MR
[12] , , and , Micro-macro limit of a nonlocal generalized aw-rascle type model. SIAM J. Appl. Math. 80 (2020) 1841–1861. | MR | DOI
[13] and , Stationary wave profiles for nonlocal particle models of traffic flow on rough roads. NoDEA Nonlinear Differ. Equ. Appl. 26 (2019) 53. | MR | DOI
[14] , , and , A non-local macroscopic model for traffic flow. ESAIM: M2AN 55 (2021) 689–711. | MR | Zbl | Numdam | DOI
[15] and , Vanishing viscosity for traffic on networks. SIAM J. Math. Anal. 42 (2010) 1761–1783. | MR | Zbl | DOI
[16] , and , Traffic flow on a road network. SIAM J. Math. Anal. 36 (2005) 1862–1886. | MR | Zbl | DOI
[17] , , , and , A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels. Preprint (2020). | arXiv
[18] and , Well posedness for multilane traffic models. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006) 291–301. | MR | Zbl | DOI
[19] , and , On the modelling and management of traffic. ESAIM: M2AN 45 (2011) 853–872. | MR | Zbl | Numdam | DOI
[20] , , and , Local limit of nonlocal traffic models: convergence results and total variation blow-up, in Annales de l’Institut Henri Poincaré C, Analyse non linéaire. Elsevier (2020). | MR | Zbl | Numdam
[21] , , and , Representation of capacity drop at a road merge via point constraints in a first order traffic model. ESAIM: M2AN 53 (2019) 1–34. | MR | Zbl | Numdam | DOI
[22] and , Maximum principle satisfying cweno schemes for nonlocal conservation laws. SIAM J. Sci. Comput. 41 (2019) A973–A988. | MR | DOI
[23] , and , Nonlocal approaches for multilane traffic models. Preprint (2020). | arXiv | MR
[24] , and , A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Netw. Heterog. Media 13 (2018) 531–547. | MR | DOI
[25] and , Traffic flow on networks, In vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, Conservation laws models (2006). | MR | Zbl
[26] , and , Models for vehicular traffic on networks, In vol. 9 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2016). | MR
[27] and , Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Netw. Heterog. Media 11 (2016) 107–121. | MR | DOI
[28] and , A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit. Commun. Math. Sci. 15 (2017) 261–287. | MR | DOI
[29] , and , Speed limit and ramp meter control for traffic flow networks. Eng. Optim. 48 (2016) 1121–1144. | MR | DOI
[30] , and , Congestion on multilane highways. SIAM J. Appl. Math. 63 (2003) 818–833. | MR | Zbl | DOI
[31] , and , A macroscopic traffic model for road networks with a representation of the capacity drop phenomenon at the junctions, in Proceedings 16th IFAC World Congress, Prague, Czech Republic, July (2005). Tu-M01-TP/3.
[32] and , Modeling and simulation of multilane traffic flow. Phys. Rev. E 55 (1997) 5498. | DOI
[33] and , Modeling, simulation, and optimization of traffic flow networks. SIAM J. Sci. Comput. 25 (2003) 1066–1087. | MR | Zbl | DOI
[34] and , A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26 (1995) 999–1017. | MR | Zbl | DOI
[35] and , Models for dense multilane vehicular traffic. SIAM J. Math. Anal. 51 (2019) 3694–3713. | MR | DOI
[36] and , Existence, uniqueness and regularity results on nonlocal balance laws. J. Differ. Equ. 263 (2017) 4023–4069. | MR | DOI
[37] and , Nonlocal conservation laws with time delay. Nonlinear Differ. Equ. Appl. NoDEA 26 (2019) 54. | MR | DOI
[38] and , On approximation of local conservation laws by nonlocal conservation laws. J. Math. Anal. Appl. 475 (2019) 1927–1955. | MR | DOI
[39] , and , Nonlocal scalar conservation laws on bounded domains and applications in traffic flow. SIAM J. Math. Anal. 50 (2018) 6271–6306. | MR | DOI
[40] , and , Capacity drop and traffic control for a second order traffic model. Netw. Heterog. Media 12 (2017) 663–681. | MR | DOI
[41] and , On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A. 229 (1955) 317–345. | MR | Zbl | DOI
[42] , and , Lane changing models: a critical review. Transp. Lett. 2 (2010) 157–173. | DOI
[43] , , , , and , Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering. J. Optim. Theory Appl. 167 (2015) 733–760. | MR | DOI
[44] , Shock waves on the highway. Oper. Res. 4 (1956) 42–51. | MR | DOI
[45] and , Traveling waves for nonlocal models of traffic flow. Discrete Contin. Dyn. Syst. 39 (2019) 4001–4040. | MR | DOI
[46] , An explicit finite volume algorithm for vanishing viscosity solutions on a network. Preprint (2020). | MR
[47] and , Traffic flow dynamics. Data, models and simulation, Translated by Treiber and Christian Thiemann. Springer, Heidelberg (2013). | MR | DOI
[48] , A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. B: Methodol. 36 (2002) 275–290. | DOI
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