Article Contents
Article Contents

# Comparative study of macroscopic traffic flow models at road junctions

• * Corresponding author: Elena Rossi
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.

Mathematics Subject Classification: Primary: 35L65, 90B20; Secondary: 82B21.

 Citation:

• 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] 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. Coclite, M. 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. 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. [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. 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. [22] B. Schnetzler, X. 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.

Figures(14)