# American Institute of Mathematical Sciences

November  2019, 24(11): 6189-6207. doi: 10.3934/dcdsb.2019135

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

 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

Both authors are members of the INdAM Research group GNCS

Received  May 2018 Revised  January 2019 Published  July 2019

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.

Citation: Emiliano Cristiani, Elisa Iacomini. An interface-free multi-scale multi-order model for traffic flow. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6189-6207. doi: 10.3934/dcdsb.2019135
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##### References:
Space-time trajectories of vehicles obeying to the system (1)-(6)-(7) with $N = 34$, $\alpha = 0.6$, $\Delta_{ \rm{min}} = 7.89$, $V_{ \rm{max}} = 1$, $\tau = 4.86$, $L = 314$
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
Step 1: Vehicles appear around large jumps of the macroscopic velocity (corresponding to large jumps of the macroscopic density)
Step 3: Green vehicles are leaders
Step 4: Red vehicles are going to be deactivated
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 $\Gamma_{j-1},\ \Gamma_j>0$ & $\Gamma_{j+1} = 0$, $\theta = 0$)
Test 1: a. $n = 1$, b. $n = 100$
Test 1: CPU time for the fully microscopic model and the multi-scale model
Test 2: a. $n = 1$, b. $n = 100$, c. $n = 400$, $\tau = 0.01$, d. $n = 400$, $\tau = 3$
Test 2: Fundamental diagram of the multi-scale model compared with that of the LWR model. a. $\tau = 0.01$, b. $\tau = 3$
Test 3: a. $n = 1$, b. $n = 22$, c. $n = 441$, d. $n = 926$
Test 4: a. $n = 1$, b. $n = 277$, c. $n = 1666$, d. $n = 2191$
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$\ell_{N}$
 $T$ $L$ $N_x$ $N_t$ $\tau$ $\Gamma_{ \rm{max}}$ $\delta v$ $\delta t$ $\delta V$ $\alpha$ $\Delta_{ \rm{min}}$ 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$\ell_{N}$