Two algorithms for a fully coupled and consistently macroscopic PDE-ODE system modeling a moving bottleneck on a road

Gabriella Bretti; Emiliano Cristiani; Corrado Lattanzio; Amelio Maurizi; Benedetto Piccoli; Gabriella Bretti; Emiliano Cristiani; Corrado Lattanzio; Amelio Maurizi; Benedetto Piccoli

doi:10.3934/Mine.2018.1.55

Mathematics in Engineering

2019, Volume 1, Issue 1: 55-83. doi: 10.3934/Mine.2018.1.55

Previous Article Next Article

Research article

Two algorithms for a fully coupled and consistently macroscopic PDE-ODE system modeling a moving bottleneck on a road

1.
Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Via deiTaurini 19, 00185 - Rome, Italy
2.
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studidell’Aquila, Via Vetoio, 67100 - Coppito (AQ), Italy
3.
Department of Mathematical Sciences, Rutgers University, 311 N 5th Street, Camden, NJ 08102,USA

Received: 19 June 2018 Accepted: 25 July 2018 Published: 14 September 2018

In this paper we propose two numerical algorithms to solve a coupled PDE-ODE system which models a slow vehicle (bottleneck) moving on a road together with other cars. The resulting system is fully coupled because the dynamics of the slow vehicle depends on the density of cars and, at the same time, it causes a capacity drop in the road, thus limiting the car flux. The first algorithm, based on the Wave Front Tracking method, is suitable for theoretical investigations and convergence results. The second one, based on the Godunov scheme, is used for numerical simulations. The case of multiple bottlenecks is also investigated.
- conservation laws,
- discontinuous ODEs,
- fluid dynamic models,
- LWR model
Citation: Gabriella Bretti, Emiliano Cristiani, Corrado Lattanzio, Amelio Maurizi, Benedetto Piccoli. Two algorithms for a fully coupled and consistently macroscopic PDE-ODE system modeling a moving bottleneck on a road[J]. Mathematics in Engineering, 2019, 1(1): 55-83. doi: 10.3934/Mine.2018.1.55

Related Papers:

Abstract

In this paper we propose two numerical algorithms to solve a coupled PDE-ODE system which models a slow vehicle (bottleneck) moving on a road together with other cars. The resulting system is fully coupled because the dynamics of the slow vehicle depends on the density of cars and, at the same time, it causes a capacity drop in the road, thus limiting the car flux. The first algorithm, based on the Wave Front Tracking method, is suitable for theoretical investigations and convergence results. The second one, based on the Godunov scheme, is used for numerical simulations. The case of multiple bottlenecks is also investigated.

References

[1]	Andreainov B, Goatin P and Seguin N (2010) Finite volume scheme for locally constrained conservation laws. Numer Math 115: 609–645. doi: 10.1007/s00211-009-0286-7
[2]	Aubin JP and Cellina A (1984) Differential inclusions. Volume 264 of Grundlehren Math. Wiss., Springer-Verlag, Berlin.
[3]	Aw A and Rascle M (2000) Resurrection of second order models of traffic flow. SIAM J Appl Math 60: 916–938. doi: 10.1137/S0036139997332099
[4]	Benzoni-Gavage S and Colombo RM (2003) An n-populations model for traffic flow. Eur J Appl Math 14: 587–612. doi: 10.1017/S0956792503005266
[5]	Bressan A (1993) A contractive metric for systems of conservation laws with coinciding shock and rarefaction curves. J Differ Equations 106: 332–366. doi: 10.1006/jdeq.1993.1111
[6]	Bressan A (2000) Hyperbolic Systems of Conservation Laws – The One-dimensional Cauchy Problem, Oxford University Press.
[7]	Bressan A, Crasta G and Piccoli B (2000)Well Posedness of the Cauchy Problem for n x n systems of conservation laws. American Mathematical Soc Memoir 694.
[8]	Bressan A and Marson A (1995) A Variational Calculus for Discontinuous Solutions of Systems of Consevation Laws. Commun Part Diff Eq 20: 1491–1552. doi: 10.1080/03605309508821142
[9]	Bressan A and LeFloch PG (1999) Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws. Indiana U Math J 48: 43–84.
[10]	Bretti G, Natalini R and Piccoli B (2006) Numerical approximations of a traffic flow model on networks. Netw Heterog Media 1: 57–84. doi: 10.3934/nhm.2006.1.57
[11]	Bretti G and Piccoli B (2008) A tracking algorithm for car paths on road networks. SIAM J Appl Dyn Syst 7: 510–531. doi: 10.1137/070697768
[12]	Bretti G, Briani M and Cristiani E (2014) An easy-to-use algorithm for simulating traffic flow on networks: numerical experiments. Discrete Contin Dyn Syst Ser S 7: 379–394. doi: 10.3934/dcdss.2014.7.379
[13]	Briani M and Cristiani E (2014) An easy-to-use algorithm for simulating traffic flow on networks: theoretical study. Netw Heterog Media 9: 519–552. doi: 10.3934/nhm.2014.9.519
[14]	Chitour Y and Piccoli B (2005) traffic circles and timing of traffic lights for cars flow. Discrete Contin Dyn Syst Ser B 5: 599–630. doi: 10.3934/dcdsb.2005.5.599
[15]	Coclite GM, Garavello M and Piccoli B (2005) traffic flow on a road network. SIAM J Math Anal 36: 1862–1886. doi: 10.1137/S0036141004402683
[16]	Colombo RM (2002) Hyperbolic phase transitions in traffic flow. SIAM J Appl Math 63: 708–721.
[17]	Colombo RM and Goatin P (2007) A well posed conservation law with a variable unilateral constraint. J Differ Equations 234: 654–675. doi: 10.1016/j.jde.2006.10.014
[18]	Colombo RM and Marcellini F (2016) A traffic model aware of real time data. Math Mod Meth Appl S 26: 445–467. doi: 10.1142/S0218202516500081
[19]	Colombo RM and Marson A (2003) Conservation laws and O.D.E.s. A traffic problem. Hyperbolic Problems: Theory, Numerics, Applications, Springer-Verlag, Berlin, Heidelberg, 455–461.
[20]	Delle Monache ML and Goatin P (2014) A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete Contin Dyn Syst Ser S 7: 435–447. doi: 10.3934/dcdss.2014.7.435
[21]	Delle Monache ML and Goatin P (2014) Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result. J Differ Equations 257: 4015–4029. doi: 10.1016/j.jde.2014.07.014
[22]	Delle Monache ML and Goatin P (2017) Stability estimates for scalar conservation laws with moving flux constraints. Netw Heterog Media 12: 245–258. doi: 10.3934/nhm.2017010
[23]	Garavello M, Natalini R, Piccoli B, et al. (2007) Conservation laws with discontinuous flux. Netw Heterog Media 2: 159–179.
[24]	Garavello M and Piccoli B (2006) traffic Flow on Networks. AIMS Series on Applied Mathematics.
[25]	Gasser I, Lattanzio C and Maurizi A (2013) Vehicular traffic flow dynamics on a bus route. Multiscale Model Sim 11: 925–942. doi: 10.1137/130906350
[26]	Gazis DC and Herman R (1992) The moving and "phantom" bottlenecks. Transport Sci 26: 223–229. doi: 10.1287/trsc.26.3.223
[27]	Godlewski E and Raviart PA (1991) Hyperbolic Systems of Conservation Laws. Mathématiques & Applications [Mathematics and Applications], 3/4. Ellipses, Paris.
[28]	Godunov SK (1959) A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Matematicheskii Sbornik 47: 271–290.
[29]	Herty M, Lebacque JP and Moutari S (2009) A novel model for intersections of vehicular traffic flow. Netw Heterog Media 4: 813–826. doi: 10.3934/nhm.2009.4.813
[30]	Holden H and Risebro NH (1995) A mathematical model of traffic flow on a network of unidirectional roads. SIAM J Math Anal 26: 999–1017. doi: 10.1137/S0036141093243289
[31]	Holden H and Risebro NH (2002) Front Tracking for Hyperbolic Conservation Laws. Vol 152. Springer Series on Applied Mathematical Sciences.
[32]	Lattanzio C, Maurizi A and Piccoli B (2011) Moving bottlenecks in car traffic flow: a PDE-ODE coupled model. SIAM J Math Anal 43: 50–67. doi: 10.1137/090767224
[33]	Lattanzio C, Maurizi A and Piccoli B (2010) Modeling and simulation of vehicular traffic flow with moving bottlenecks, In: Pistella F and Spitaleri RM (eds.), Proceedings of MASCOT09, Vol 15 of IMACS Series in Computational and Applied Mathematics, Rome, 181–190.
[34]	Laval JA and Daganzo CF (2006) Lane-changing in traffic streams. Transport Res B-Meth 40: 251–264. doi: 10.1016/j.trb.2005.04.003
[35]	Lebacque JP (1996) The Godunov scheme and what it means for first order flow models. Transportation and traffic Theory. Proceedings of the 13th International Symposium on Transportation and traffic Theory, Pergamon, Oxford, 647–677.
[36]	Lebacque JP, Lesort JB and Giorgi F (1998) Introducing buses into first order macroscopic traffic flow models. Transport Res Rec 1644: 70–79. doi: 10.3141/1644-08
[37]	Liard T and Piccoli B (2018) Well-posedness for scalar conservation laws with moving flux constraints. Preprint ArXiv: 1801.04814.
[38]	Lighthill MJ and Whitham GB (1955) On kinetic waves. II. Theory of traffic flows on long crowded roads. Proc Roy Soc London Ser A 229: 317–345.
[39]	Newell GF (1988) A moving bottleneck. Transport Res B-Meth 32: 531–537.
[40]	Piacentini G, Goatin P and Ferrara A (2018) traffic control via moving bottleneck of coordinated vehicles. Proceedings of the 15th IFAC Symposium on Control in Transportation Systems, Savona (Italy), 51: 13–18.
[41]	Richards PI (1956) Shock waves on the highway. Oper Res 4: 42–51. doi: 10.1287/opre.4.1.42
[42]	Villa S, Goatin P and Chalons C (2017) Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model. Discrete Cont Dyn-B 22: 3921–3952. doi: 10.3934/dcdsb.2017202
[43]	Wong GCK and Wong SC (2002) A multi-class traffic flow model: an extension of LWR model with heterogeneous drivers. Transport Res A-Pol 36: 827–841.

Reader Comments

Your name:*

Email:*
© 2019 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)