June  2020, 15(2): 261-279. doi: 10.3934/nhm.2020012

Comparative study of macroscopic traffic flow models at road junctions

1. 

Inria Sophia Antipolis - Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 2004 route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France

2. 

Università degli Studi di Milano-Bicocca, Dipartimento di Matematica e Applicazioni, via R. Cozzi 55, 20126 Milano, Italy

* Corresponding author: Elena Rossi

Received  February 2020 Published  June 2020 Early access  April 2020

Fund Project: The second author is a member of INdAM-GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni)

We qualitatively compare the solutions of a multilane model with those produced by the classical Lighthill-Whitham-Richards equation with suitable coupling conditions at simple road junctions. The numerical simulations are based on the Godunov and upwind schemes. Several tests illustrate the models' behaviour in different realistic situations.

Citation: Paola Goatin, Elena Rossi. Comparative study of macroscopic traffic flow models at road junctions. Networks and Heterogeneous Media, 2020, 15 (2) : 261-279. doi: 10.3934/nhm.2020012
References:
[1]

Adimurthi and G. D. V. Gowda, Conservation law with discontinuous flux, J. Math. Kyoto Univ., 43 (2003), 27–70. doi: 10.1215/kjm/1250283740.

[2]

J. J. Adimurthi and G. D. Veerappa Gowda, Godunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal., 42 (2004), 179-208.  doi: 10.1137/S003614290139562X.

[3]

M. S. Adimurthi and G. D. V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.  doi: 10.1142/S0219891605000622.

[4]

G. M. CocliteM. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.  doi: 10.1137/S0036141004402683.

[5]

E. Cristiani and F. S. Priuli, A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks, Netw. Heterog. Media, 10 (2015), 857-876.  doi: 10.3934/nhm.2015.10.857.

[6]

S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26 (1995), 1425-1451.  doi: 10.1137/S0036141093242533.

[7]

S. Diehl, Scalar conservation laws with discontinuous flux function. I. The viscous profile condition, Comm. Math. Phys., 176 (1996), 23-44.  doi: 10.1007/BF02099361.

[8]

A. Festa and P. Goatin, Modeling the impact of on-line navigation devices in traffic flows, 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France (2019), 323–328. doi: 10.1109/CDC40024.2019.9030208.

[9]

M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, AIMS Series on Applied Mathematics, 9, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016.

[10]

M. Garavello and B. Piccoli, Source-destination flow on a road network, Commun. Math. Sci., 3 (2005), 261-283.  doi: 10.4310/CMS.2005.v3.n3.a1.

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[12]

T. Gimse and N. H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), 635-648.  doi: 10.1137/0523032.

[13]

P. GoatinS. Göttlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Eng. Optim., 48 (2016), 1121-1144.  doi: 10.1080/0305215X.2015.1097099.

[14]

P. Goatin and E. Rossi, A multiLane macroscopic traffic flow model for simple networks, SIAM J. Appl. Math., 79 (2019), 1967-1989.  doi: 10.1137/19M1254386.

[15]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.  doi: 10.1137/S0036141093243289.

[16]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Applied Mathematical Sciences, 152, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2.

[17]

H. Holden and N. H. Risebro, Models for dense multilane vehicular traffic, SIAM J. Math. Anal., 51 (2019), 3694-3713.  doi: 10.1137/19M124318X.

[18]

K. H. Karlsen, N. H. Risebro and J. D. Towers, $L^1$stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk., (2003), 1–49.

[19]

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: 10.1098/rspa.1955.0089.

[20]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[21]

S. SamaranayakeW. KricheneJ. ReillyM. L. D. MonacheP. Goatin and A. Bayen, Discrete-time system optimal dynamic traffic assignment (SO-DTA) with partial control for physical queuing networks, Transportation Science, 52 (2018), 982-1001.  doi: 10.1287/trsc.2017.0800.

[22]

B. SchnetzlerX. Louis and J.-P. Lebacque, A multilane junction model, TRANSPORTMETRICA, 8 (2012), 243-260.  doi: 10.1080/18128601003752452.

[23]

J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.  doi: 10.1137/S0036142999363668.

show all references

References:
[1]

Adimurthi and G. D. V. Gowda, Conservation law with discontinuous flux, J. Math. Kyoto Univ., 43 (2003), 27–70. doi: 10.1215/kjm/1250283740.

[2]

J. J. Adimurthi and G. D. Veerappa Gowda, Godunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal., 42 (2004), 179-208.  doi: 10.1137/S003614290139562X.

[3]

M. S. Adimurthi and G. D. V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.  doi: 10.1142/S0219891605000622.

[4]

G. M. CocliteM. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.  doi: 10.1137/S0036141004402683.

[5]

E. Cristiani and F. S. Priuli, A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks, Netw. Heterog. Media, 10 (2015), 857-876.  doi: 10.3934/nhm.2015.10.857.

[6]

S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26 (1995), 1425-1451.  doi: 10.1137/S0036141093242533.

[7]

S. Diehl, Scalar conservation laws with discontinuous flux function. I. The viscous profile condition, Comm. Math. Phys., 176 (1996), 23-44.  doi: 10.1007/BF02099361.

[8]

A. Festa and P. Goatin, Modeling the impact of on-line navigation devices in traffic flows, 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France (2019), 323–328. doi: 10.1109/CDC40024.2019.9030208.

[9]

M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, AIMS Series on Applied Mathematics, 9, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016.

[10]

M. Garavello and B. Piccoli, Source-destination flow on a road network, Commun. Math. Sci., 3 (2005), 261-283.  doi: 10.4310/CMS.2005.v3.n3.a1.

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[12]

T. Gimse and N. H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), 635-648.  doi: 10.1137/0523032.

[13]

P. GoatinS. Göttlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Eng. Optim., 48 (2016), 1121-1144.  doi: 10.1080/0305215X.2015.1097099.

[14]

P. Goatin and E. Rossi, A multiLane macroscopic traffic flow model for simple networks, SIAM J. Appl. Math., 79 (2019), 1967-1989.  doi: 10.1137/19M1254386.

[15]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.  doi: 10.1137/S0036141093243289.

[16]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Applied Mathematical Sciences, 152, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2.

[17]

H. Holden and N. H. Risebro, Models for dense multilane vehicular traffic, SIAM J. Math. Anal., 51 (2019), 3694-3713.  doi: 10.1137/19M124318X.

[18]

K. H. Karlsen, N. H. Risebro and J. D. Towers, $L^1$stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk., (2003), 1–49.

[19]

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: 10.1098/rspa.1955.0089.

[20]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[21]

S. SamaranayakeW. KricheneJ. ReillyM. L. D. MonacheP. Goatin and A. Bayen, Discrete-time system optimal dynamic traffic assignment (SO-DTA) with partial control for physical queuing networks, Transportation Science, 52 (2018), 982-1001.  doi: 10.1287/trsc.2017.0800.

[22]

B. SchnetzlerX. Louis and J.-P. Lebacque, A multilane junction model, TRANSPORTMETRICA, 8 (2012), 243-260.  doi: 10.1080/18128601003752452.

[23]

J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.  doi: 10.1137/S0036142999363668.

Figure 1.  The demand (left) and supply (right) function for $ f_I $ as in (2). In both pictures, the dashed line represents $ f_I (u) $
Figure 2.  The junction types considered in this work for the LWR model
Figure 3.  Scheme of the 1-to-1 junction with 2 lanes on the incoming road and 3 lanes on the outgoing one
Figure 4.  Flux functions $ f_\ell $ and $ f_r $ related to the LWR model, for $ M_\ell = 2 $, $ M_r = 3 $, $ V_\ell = 1.5 $ and $ V_r = 1 $. The point $ \check u $ is such that $ f_\ell(\check u) = f_r (\check u) $. The value $ U $ corresponds to the left trace at $ x = 0 $ of the sum of the solutions of the multilane model on the two incoming lanes, and $ f_\ell(U) $ is the corresponding value of the flux
Figure 5.  The dashed blue line corresponds to the multilane model (12)–(16)–(19): on the left, it is the sum of its solutions, on the right it is the average. The dash-dotted orange line corresponds to the solution to the LWR model (1) obtained via the Godunov type scheme; the dotted green line is the solution to the LWR model (1) obtained through the upwind scheme. Here: $ V_\ell = 1.5 $, $ V_r = 1 $, initial datum (25)
Figure 6.  Left: Flux functions $ f_\ell $ and $ f_r $ related to the LWR model when comparing it to the multilane model in the form of the average of the densities on the various lanes: $ V_\ell = 1.5 $ and $ V_r = 1 $, in both cases the maximal density is $ 1 $. The orange line represents the solution to the Riemann problem with initial datum $ \rho_o (x) = 0.5 $. Right: flux functions related to the multilane model for the incoming ($ f_\ell $) and the outgoing ($ f_r $) lanes. The magenta line represents the solution on lane 1, with initial datum $ \rho_{o,1} = 0.6 $; the dotted blue line corresponds to the solution on lane 2, with initial datum $ \rho_{o,2} = 0.4 $
Figure 7.  Flux functions $ f_\ell $ and $ f_r $ related to the LWR model, for $ M_\ell = 2 $, $ M_r = 3 $, $ V_\ell = 1 $ and $ V_r = 1.5 $. The point $ \tilde u $ is such that $ f_r(\tilde u) = f_\ell (1) $
Figure 8.  The dashed blue line is the sum of the solutions to the multilane model (12)–(16)–(19); the dash-dotted orange line corresponds to the solution to the LWR model (1) obtained via the Godunov type scheme; the dotted green line is the solution to the LWR model (1) obtained through the upwind scheme. Here: $ V_\ell = 1 $, $ V_r = 1.5 $, initial datum (25)
Figure 9.  Scheme of the 2-to-1 junction with 2 lanes on each incoming road and 2 lanes on the outgoing one
Figure 10.  In each picture, the dashed blue line is the sum of the solutions to the multilane model (12)–(16)–(19): from left to right, lanes 1 and 2; lanes 3 and 4; lanes 2 and 3. The dash-dotted orange line corresponds to the solution to the LWR model (1), obtained through a Godunov type scheme, with priorities $ \left(1/2, 1/2\right) $: from left to right, incoming roads $ a $, $ b $, outgoing road $ c $. The initial data are given in (26) and (27) respectively. In each lane we set $ V = 1.5 $
Figure 11.  Solution to the multilane model (12)–(16)–(19) at time $ t = 1 $, with initial data (26) and $ V = 1.5 $ on each lane
Figure 12.  Scheme of the 1-to-2 junction (diverging), with 2 lanes on the incoming road and 1 lane on both outgoing roads
Figure 13.  In each picture, the dashed blue line corresponds to the solutions to the multilane multi-population mode (29), and in particular to their sum on $ x<0 $. The dotted orange line corresponds to the solution of the LWR mode (1) with non-FIFO rule, while the dash-dotted green line is the solution to the LWR mode (1) with FIFO rule. The initial data is (31), while $ V_\ell = 1.5 $, $ V_r = 2 $ and $ \alpha = 0.4 $
Figure 14.  In each picture, the dashed blue line corresponds to the solutions to the multilane multi-population mode (29), and in particular to their sum on $ x<0 $. The dotted orange line corresponds to the solution of the LWR mode (1) with non-FIFO rule, while the dash-dotted green line is the solution to the LWR mode (1) with FIFO rule. The initial data is (32), while $ V_\ell = 1.5 $, $ V_r = 2 $ and $ \alpha = 0.4 $
[1]

Michael Herty, Reinhard Illner. Analytical and numerical investigations of refined macroscopic traffic flow models. Kinetic and Related Models, 2010, 3 (2) : 311-333. doi: 10.3934/krm.2010.3.311

[2]

Paola Goatin. Traffic flow models with phase transitions on road networks. Networks and Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287

[3]

Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 283-313. doi: 10.3934/dcdss.2021005

[4]

Felisia Angela Chiarello, Paola Goatin. Non-local multi-class traffic flow models. Networks and Heterogeneous Media, 2019, 14 (2) : 371-387. doi: 10.3934/nhm.2019015

[5]

Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431

[6]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks and Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255

[7]

Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161

[8]

Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks and Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773

[9]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic and Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165

[10]

Raimund Bürger, Kenneth H. Karlsen, John D. Towers. On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Networks and Heterogeneous Media, 2010, 5 (3) : 461-485. doi: 10.3934/nhm.2010.5.461

[11]

Emiliano Cristiani, Smita Sahu. On the micro-to-macro limit for first-order traffic flow models on networks. Networks and Heterogeneous Media, 2016, 11 (3) : 395-413. doi: 10.3934/nhm.2016002

[12]

Robert Carlson. Myopic models of population dynamics on infinite networks. Networks and Heterogeneous Media, 2014, 9 (3) : 477-499. doi: 10.3934/nhm.2014.9.477

[13]

N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59

[14]

Bertrand Haut, Georges Bastin. A second order model of road junctions in fluid models of traffic networks. Networks and Heterogeneous Media, 2007, 2 (2) : 227-253. doi: 10.3934/nhm.2007.2.227

[15]

Michael Burger, Simone Göttlich, Thomas Jung. Derivation of second order traffic flow models with time delays. Networks and Heterogeneous Media, 2019, 14 (2) : 265-288. doi: 10.3934/nhm.2019011

[16]

Gabriella Puppo, Matteo Semplice, Andrea Tosin, Giuseppe Visconti. Kinetic models for traffic flow resulting in a reduced space of microscopic velocities. Kinetic and Related Models, 2017, 10 (3) : 823-854. doi: 10.3934/krm.2017033

[17]

Michael Herty, Lorenzo Pareschi, Mohammed Seaïd. Enskog-like discrete velocity models for vehicular traffic flow. Networks and Heterogeneous Media, 2007, 2 (3) : 481-496. doi: 10.3934/nhm.2007.2.481

[18]

Simone Göttlich, Oliver Kolb, Sebastian Kühn. Optimization for a special class of traffic flow models: Combinatorial and continuous approaches. Networks and Heterogeneous Media, 2014, 9 (2) : 315-334. doi: 10.3934/nhm.2014.9.315

[19]

Sharif E. Guseynov, Shirmail G. Bagirov. Distributed mathematical models of undetermined "without preference" motion of traffic flow. Conference Publications, 2011, 2011 (Special) : 589-600. doi: 10.3934/proc.2011.2011.589

[20]

Matteo Piu, Gabriella Puppo. Stability analysis of microscopic models for traffic flow with lane changing. Networks and Heterogeneous Media, 2022, 17 (4) : 495-518. doi: 10.3934/nhm.2022006

2021 Impact Factor: 1.41

Metrics

  • PDF downloads (312)
  • HTML views (183)
  • Cited by (0)

Other articles
by authors

[Back to Top]