An interface-free multi-scale multi-order model for traffic flow

Emiliano Cristiani; Elisa Iacomini

doi:10.3934/dcdsb.2019135

Article Contents

2019, Volume 24, Issue 11: 6189-6207. Doi: 10.3934/dcdsb.2019135

This issue Previous Article Linearized stability for abstract functional differential equations subject to state-dependent delays with applications Next Article Portfolio optimization and model predictive control: A kinetic approach

An interface-free multi-scale multi-order model for traffic flow

Emiliano Cristiani^1, , and
Elisa Iacomini^2,

1.
Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Rome, Italy
2.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza, Università di Roma, Via Scarpa 16, 00161 Rome, Italy

^*Corresponding author
^*Corresponding author

Both authors are members of the INdAM Research group GNCS

Received: May 2018

Revised: January 2019

Early access: July 2019

Published: August 2019

Abstract / Introduction Full Text(HTML) Figure(12) / Table(1) Related Papers Cited by

Abstract

In this paper we present a new multi-scale method for reproducing traffic flow which couples a first-order macroscopic model with a second-order microscopic model, avoiding any interface or boundary conditions between them. The multi-scale model is characterized by the fact that microscopic and macroscopic descriptions are not spatially separated. On the contrary, the macro-scale is always active while the micro-scale is activated only if needed by the traffic conditions. The Euler-Godunov scheme associated to the model is conservative and it is able to reproduce typical traffic phenomena like stop & go waves.

Keywords:

Mathematics Subject Classification: Primary: 65C20, 35M33; Secondary: 35L65.

Citation:

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Figure 1. Space-time trajectories of vehicles obeying to the system (1)-(6)-(7) with $ N = 34 $, $ \alpha = 0.6 $, $ \Delta_{ \rm{min}} = 7.89 $, $ V_{ \rm{max}} = 1 $, $ \tau = 4.86 $, $ L = 314 $

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Figure 2. Zoom of the trajectories shown in Fig. 1 around initial time. It is well visible the emergence of the stop & go wave from the interaction between the first and the last vehicle

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Figure 3. Step 1: Vehicles appear around large jumps of the macroscopic velocity (corresponding to large jumps of the macroscopic density)

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Figure 4. Step 3: Green vehicles are leaders

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Figure 5. Step 4: Red vehicles are going to be deactivated

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Figure 6. Step 9: Update of density $ \rho_j $ using microscopic flux on the left boundary and macroscopic flux on the right boundary of the cell $ j $ (case $ \Gamma_{j-1},\ \Gamma_j>0 $ & $ \Gamma_{j+1} = 0 $, $ \theta = 0 $)

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Figure 7. Test 1: a. $ n = 1 $, b. $ n = 100 $

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Figure 8. Test 1: CPU time for the fully microscopic model and the multi-scale model

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Figure 9. Test 2: a. $ n = 1 $, b. $ n = 100 $, c. $ n = 400 $, $ \tau = 0.01 $, d. $ n = 400 $, $ \tau = 3 $

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Figure 10. Test 2: Fundamental diagram of the multi-scale model compared with that of the LWR model. a. $ \tau = 0.01 $, b. $ \tau = 3 $

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Figure 11. Test 3: a. $ n = 1 $, b. $ n = 22 $, c. $ n = 441 $, d. $ n = 926 $

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Figure 12. Test 4: a. $ n = 1 $, b. $ n = 277 $, c. $ n = 1666 $, d. $ n = 2191 $

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Table 1. Model and algorithm parameters used for the numerical tests

	$ T $	$ L $	$ N_x $	$ N_t $	$ \tau $	$ \Gamma_{ \rm{max}} $	$ \delta v $	$ \delta t $	$ \delta V $	$ \alpha $	$ \Delta_{ \rm{min}} $
T1	3	20	100	300	0.01	20	0.08	15$ \Delta t $	0.3	-	-
T2	3	20	100	600	0.01Ƀ3	30	0.1	15$ \Delta t $	0.5	-	-
T3	12	20	100	1200	0.1	30	0.1	30$ \Delta t $	0.2	-	-
T4	500	314	35	4000	4.86	16	0.3	250$ \Delta t $	0.07	0.47	2.6$ \ell_{N} $

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