The Riemann problem for a Two-Phase model for road traffic with fixed or moving constraints

Francesca Marcellini; Francesca Marcellini

doi:10.3934/mbe.2020062

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1218-1232. doi: 10.3934/mbe.2020062

Previous Article Next Article

Research article Special Issues

The Riemann problem for a Two-Phase model for road traffic with fixed or moving constraints

Francesca Marcellini ^,

INdAM Unit, Department of Information Engineering, University of Brescia, Via Branze 38, I-25123, Brescia, Italy

Received: 09 July 2019 Accepted: 05 November 2019 Published: 15 November 2019

We define two Riemann solvers for the Two-Phase traffic model proposed in ^[1], given by a system of two conservation laws with Lipschitz continuous flow, under fixed and moving constraints. From the traffic point of view this situation corresponds to the study of vehicular flow with fixed constraints as, for instance, a traffic light, a toll gate or a construction site. On the other hand, the presence of a slow moving large vehicle, like a bus, corresponds to the case of a moving constraint. In the latter case, we have to consider a mixed system where the conservation laws are coupled with an ordinary differential equation describing the trajectory of the large vehicle.
- hyperbolic systems of conservation laws,
- continuum traffic models,
- evolution problems,
- unilateral flux constraints,
- moving bottlenecks
Citation: Francesca Marcellini. The Riemann problem for a Two-Phase model for road traffic with fixed or moving constraints[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1218-1232. doi: 10.3934/mbe.2020062

Related Papers:

Abstract

We define two Riemann solvers for the Two-Phase traffic model proposed in ^[1], given by a system of two conservation laws with Lipschitz continuous flow, under fixed and moving constraints. From the traffic point of view this situation corresponds to the study of vehicular flow with fixed constraints as, for instance, a traffic light, a toll gate or a construction site. On the other hand, the presence of a slow moving large vehicle, like a bus, corresponds to the case of a moving constraint. In the latter case, we have to consider a mixed system where the conservation laws are coupled with an ordinary differential equation describing the trajectory of the large vehicle.

References

[1]	R. M. Colombo, F. Marcellini, M. Rascle, A 2-phase traffic model based on a speed bound. SIAM J. Appl. Math., 70 (2010), 2652-2666.
[2]	M. J. Lighthill, 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.
[3]	P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.
[4]	A. Aw, M. Rascle, Resurrection of "second order" models of traffic flow. SIAM J. Appl. Math., 60 (electronic) (2000), 916-938.
[5]	H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transport. Res. B-Meth., 36 (2002), 275-290.
[6]	S. Fan, M. Herty, B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-RascleZhang model, Netw. Heterog. Media, 9 (2014), 239-268.
[7]	S. Fan, Y. Sun, B. Piccoli, B. Seibold, D. B. Work, A Collapsed Generalized Aw-Rascle-Zhang Model and Its Model Accuracy, arXiv preprint arXiv:1702.03624, 2017.
[8]	S. Blandin, D. Work, P. Goatin, B. Piccoli, A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127.
[9]	R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.
[10]	R. M. Colombo, F. Marcellini, A mixed ODE-PDE model for vehicular traffic, Math. Method. Appl. Sci., 38 (2015), 1292-1302.
[11]	E. Dal Santo, M. D. Rosini, N. Dymski. The Riemann problem for a general phase transition model on networks. In Theory, numerics and applications of hyperbolic problems. I, volume 236 of Springer Proc. Math. Stat., Springer, Cham (2018), 445-457.
[12]	P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Model., 44 (2006), 287-303.
[13]	J. P. Lebacque, X. Louis, S. Mammar, B. Schnetzlera, H. Haj-Salem, Modélisation du trafic autoroutier au second ordre, CR Math., 346 (2008), 1203-1206.
[14]	F. Marcellini, Free-congested and micro-macro descriptions of traffic flow, Discrete Cont. Dyn-S, 7 (2014), 543-556.
[15]	F. Marcellini, Existence of solutions to a boundary value problem for a phase transition traffic model, Networks and Heterogeneous Media, 12 (2017), 259-275.
[16]	S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.
[17]	P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., (1973). Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11.
[18]	R. M. Colombo, M. D. Rosini. Pedestrian flows and non-classical shocks. Math. Meth. Appl. S., 28 (2005), 1553-1567.
[19]	R. M. Colombo, P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differ. Equations, 234 (2007), 654-675.
[20]	B. Andreianov, C. Donadello, M. D. Rosini, A second-order model for vehicular traffics with local point constraints on the flow. Math. Models Meth. Appl. S., 26 (2016), 751-802.
[21]	R. M. Colombo, P. Goatin, M. D. Rosini, On the modelling and management of traffic, ESAIM Math. Model. Numer. Anal., 45 (2011), 853-872.
[22]	M. Garavello, P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648.
[23]	M. Garavello, S. Villa, The Cauchy problem for the Aw-Rascle-Zhang traffic model with locally constrained flow, J. Hyperbol. Differ. Eq., 14 (2017), 393-414.
[24]	M. Benyahia, C. Donadello, N. Dymski, M. D. Rosini, An existence result for a constrained twophase transition model with metastable phase for vehicular traffic, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 48, 42.
[25]	M. Benyahia, M. D. Rosini, A macroscopic traffic model with phase transitions and local point constraints on the flow, Netw. Heterog. Media, 12 (2017), 297-317.
[26]	E. Dal Santo, M. D. Rosini, N. Dymski, M. Benyahia, General phase transition models for vehicular traffic with point constraints on the flow, Math. Methods Appl. Sci., 40 (2017), 6623-6641.
[27]	M. L. Delle Monache, P. Goatin, Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result, J. Differential Equations, 257 (2014), 4015-4029.
[28]	S. Villa, P. Goatin, C. Chalons, Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model. Discrete Contin. Dyn-B, 22 (2017), 3921-3952.
[29]	T. Liard, F. Marcellini, B. Piccoli, The Riemann problem for the Garz model with a moving constraint. Hyperbolic Problems: Theory, Numerics, Applications, AIMS, to appear, (2019).
[30]	B. Andreianov, C. Donadello, U. Razafison, M. D. Rosini, One-dimensional conservation laws with nonlocal point constraints on the flux. In Crowd dynamics, Volume 1. Theory, models, and safety problems, Cham: Birkhäuser (2018), 103-135.
[31]	M. Garavello, F. Marcellini, The Godunov method for a 2-phase model, Commun. Appl. Ind. Math., 8 (2017), 149-164.
[32]	M. Garavello, F. Marcellini, The Riemann problem at a junction for a phase-transition traffic model, Discrete Cont. Dyn-A, 37 (2017), 5191-5209.
[33]	M. Garavello, B. Piccoli, Traffic flow on networks, volume 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, (2006). Conservation laws models.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)