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 $\alpha = 0.4$

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