Comparative study of macroscopic traffic flow models at road junctions

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 $