# American Institute of Mathematical Sciences

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  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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.  doi: 10.1137/S0036141004402683.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] P. Goatin, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar [21] S. Samaranayake, W. Krichene, J. Reilly, M. L. D. Monache, P. 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.  Google Scholar [22] B. Schnetzler, X. Louis and J.-P. Lebacque, A multilane junction model, TRANSPORTMETRICA, 8 (2012), 243-260.  doi: 10.1080/18128601003752452.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.  doi: 10.1137/S0036141004402683.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] P. Goatin, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar [21] S. Samaranayake, W. Krichene, J. Reilly, M. L. D. Monache, P. 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.  Google Scholar [22] B. Schnetzler, X. Louis and J.-P. Lebacque, A multilane junction model, TRANSPORTMETRICA, 8 (2012), 243-260.  doi: 10.1080/18128601003752452.  Google Scholar [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.  Google Scholar
The demand (left) and supply (right) function for $f_I$ as in (2). In both pictures, the dashed line represents $f_I (u)$
The junction types considered in this work for the LWR model
Scheme of the 1-to-1 junction with 2 lanes on the incoming road and 3 lanes on the outgoing one
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
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)
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 $\rho_{o,2} = 0.4$
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 $f_r(\tilde u) = f_\ell (1)$
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)
Scheme of the 2-to-1 junction with 2 lanes on each incoming road and 2 lanes on the outgoing one
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 $V = 1.5$
Solution to the multilane model (12)–(16)–(19) at time $t = 1$, with initial data (26) and $V = 1.5$ on each lane
Scheme of the 1-to-2 junction (diverging), with 2 lanes on the incoming road and 1 lane on both outgoing roads
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 $\alpha = 0.4$
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 $\alpha = 0.4$
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