A two-dimensional multi-class traffic flow model

Caterina Balzotti; Simone Göttlich

doi:10.3934/nhm.2020034

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2021, Volume 16, Issue 1: 69-90. Doi: 10.3934/nhm.2020034

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A two-dimensional multi-class traffic flow model

Caterina Balzotti^1, , and
Simone Göttlich^2,

1.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Rome, Italy
2.
Department of Mathematics, University of Mannheim, B6 28-29, 68159 Mannheim, Germany

^* Corresponding author: Caterina Balzotti
^* Corresponding author: Caterina Balzotti

Received: May 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: March 2021

Part of this work was carried out while the first author was visiting the University of Mannheim within the program IPID4all funded by the German Academic Exchange Service (DAAD)

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Abstract

The aim of this work is to introduce a two-dimensional macroscopic traffic model for multiple populations of vehicles. Starting from the paper [21], where a two-dimensional model for a single class of vehicles is proposed, we extend the dynamics to a multi-class model leading to a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems we also present a Lax-Friedrichs type discretization scheme recovering the theoretical results by means of numerical tests. We calibrate the multi-class model with real data and compare the fitted model to the real trajectories. Finally, we test the ability of the model to simulate the overtaking of vehicles.

Keywords:

Mathematics Subject Classification: Primary: 90B20, 35L65; Secondary: 35Q91.

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Figure 1. Representation of no shocks (A) and no rarefaction waves (B)

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Figure 2. Representation of exactly one shock, where $ \mathit{R} $ denotes rarefaction waves and $ \mathit{S} $ shocks

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Figure 3. Representation of exactly one rarefaction wave, where $ \mathit{R} $ denotes rarefaction waves, $ \mathit{S} $ shocks and $ \mathit{C.D.} $ contact discontinuities

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Figure 4. Representation of two shocks and two rarefaction waves, where $ \mathit{R} $ denotes rarefaction waves and $ \mathit{S} $ shocks

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Figure 5. Numerical solutions of Riemann problems depending on the initial datum

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Figure 6. German motorway A3 structure, cf. [25]

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Figure 7. Speed-density and flux-density diagrams for the two classes related to the $ x $-direction in the first row, and to the $ y $-direction on the second row

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Figure 8. Speed-density and flux-density diagrams for the two classes defined from real data (green and blue circles) and family of speed and velocity functions related to the $ x $-direction in the first row, and to the $ y $-direction in the second row

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Figure 9. Contours of the density of cars (top) and trucks (bottom): initial condition at time $ t = 0 $ (left), simulated results at time $ t = 5\,\mathrm{s} $ (middle) and reconstructed real data at time $ t = 5\,\mathrm{s} $ (right)

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Figure 10. Error between real and numerical density of cars and trucks during 10 seconds of simulation, computed every 0.5 seconds

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Figure 11. Speed-density and flow-density diagrams for the two classes defined from real data (green and blue circles) and family of speed and flux functions defined by (19)

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Figure 12. Error between real density and numerical density of cars and trucks during 10 seconds of simulation, computed every 0.5 seconds

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Figure 13. Contours of the density of cars and truck at time $ t = 0 $ (left), $ t = T/4 $ (middle) and $ t = T $ (right). The truck does not move, Car 1 leaves the road during the simulation and Car 2 overtakes the truck and is exiting the road at time $ T $

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Figure 14. Plot of the density of cars and truck on Lane 1 (first row) and Lane 2 (second row) at time $ t = 0 $ (left), $ t = T/4 $ (middle) and $ t = T $ (right)

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