Geodesic
Multirate solver with speed and gap error control for vehicular traffic simulation
Matematičeskoe modelirovanie, Tome 30 (2018) no. 9, pp. 87-99.

Voir la notice de l'article provenant de la source Math-Net.Ru

Nowadays computer simulation of vehicular traffic on the real road network can be used as a tool for solving actual and practical problems. The microscopic approach and large number of vehicles to simulate (tens of thousands) lead to tremendous systems of ordinary differential systems. The vehicles dynamics can vary sufficiently from vehicle to vehicle. As a result the corresponding differential equations system has different time scales, which are localized over the components. In other words, the temporal variations have different time scales for different components, which are in this case vehicles speeds and distances between them (gaps). In this paper we suggest the numerical integration scheme, which exploits an individual time step for each component (microstep) within one macrostep. The microstep value of a particular system component is determined by the local temporal variation of the solution, instead of using a single step size for the whole system. This time stepping strategy is obtained both for vehicles speeds and gaps. What is more, the local error estimation for the gaps is derived one order higher than for the speeds, because drivers assess first of all the distance, not the speed. Comparison with the corresponding single-rate scheme demonstrates substantial gains in CPU times.
Keywords: numerical integration, multirate solvers, ordinary differential equations, microscopic vehicular traffic models. 
@article{MM_2018_30_9_a5,
     author = {V. V. Kurtc and I. E. Anufriev},
     title = {Multirate solver with speed and gap error control for vehicular traffic simulation},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {87--99},
     publisher = {mathdoc},
     volume = {30},
     number = {9},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2018_30_9_a5/}
}
TY  - JOUR
AU  - V. V. Kurtc
AU  - I. E. Anufriev
TI  - Multirate solver with speed and gap error control for vehicular traffic simulation
JO  - Matematičeskoe modelirovanie
PY  - 2018
SP  - 87
EP  - 99
VL  - 30
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2018_30_9_a5/
LA  - ru
ID  - MM_2018_30_9_a5
ER  - 
%0 Journal Article
%A V. V. Kurtc
%A I. E. Anufriev
%T Multirate solver with speed and gap error control for vehicular traffic simulation
%J Matematičeskoe modelirovanie
%D 2018
%P 87-99
%V 30
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2018_30_9_a5/
%G ru
%F MM_2018_30_9_a5
V. V. Kurtc; I. E. Anufriev. Multirate solver with speed and gap error control for vehicular traffic simulation. Matematičeskoe modelirovanie, Tome 30 (2018) no. 9, pp. 87-99. http://geodesic.mathdoc.fr/item/MM_2018_30_9_a5/

[1] M. Treiber, A. Kesting, Traffic Flow Dynamics, Springer, Berlin, 2013, 503 pp.

[2] A.V. Gasnikov, Vvedenie v matematicheskoe modelirovanie transportnykh potokov, MFTI, M., 2010, 362 pp.

[3] G. Orosz, R.E. Wilson, B. Krauskopf, “Global bifurcation investigation of an optimal velocity traffic model with driver reaction time”, Phys. Rev. E, 70 (2004), 026207

[4] A. Tordeux, S. Lassarre, M. Roussignol, “An adaptive time gap car-following model”, Transportation Research Part B, 44 (2010), 1115–1131

[5] C. Gear, D. Wells, “Multirate linear multistep methods”, BIT, 24:4 (1984), 484–502

[6] M. Gunther, A. Kvœrnø, P. Rentrop, “Multirate partitioned Runge-Kutta methods”, BIT, 41 (2001), 504–514

[7] V. Savcenco, W. Hundsdorfer, J. G. Verwer, “A multirate time stepping strategy for stiff ordinary differential equations”, BIT, 47 (2007), 137–155

[8] A.B. Korchak, A.V. Evdokimov, “Metod parallelnogo rascheta rasshcheplennykh sistem differentsialnykh uravnenii s kratnimi shagami”, Trudy MFTI, 2:2 (2010), 77–85

[9] V.V. Kurtc, I.E. Anufriev, “Multirate numerical scheme for large-scale vehicle traffic simulation”, Mathematical Models and Computer Simulations, 8:6 (2016), 744–751

[10] H.J. Payne, “Models of freeway traffic and control”, Mathematical Models of Public Systems, Simulation Council Proc. 28, v. 1, ed. G. A. Bekey, 1971, 51–61

[11] B.S. Kerner, P. Konhäuser, “Cluster effect in initially homogeneous traffic flow”, Phys. Rev. E, 48 (1993), 2335–2338

[12] A.A. Chechina, N.G. Churbanova, M. A. Trapeznikova, “Two-dimensional hydrodynamic model for traffic flow simulation using parallel computer systems”, Proceedings of the international conference of the numerical analysis and applied mathematics 2014, AIP Conference Proceedings, 1648, 2015, 530007

[13] R. Jiang, Q. Wu, Z. Zhu, “Full velocity difference model for a car-following theory”, Physical Review E, 64:1 (2001), 017101.1–017101.4

[14] M. Treiber, A. Kesting, D. Helbing, “Delays, inaccuracies and anticipation in microscopic traffic models”, Physica A, 360:1 (2006), 71–88

[15] I. Lubashevsky, P. Wagner, R. Manhke, “A bounded rational driver model”, European Physical Journal B, 32 (2003), 243–247

[16] A. Kvœrnø, “Stability of multirate Runge-Kutta schemes”, Int. J. Differ. Equ. Appl., 1(A) (2000), 97–105

[17] S. Skelboe, “Stability properties of backward differentiation multirate formulas”, Appl. Numer. Math., 5 (1989), 151–160

[18] Verhoeven et al., Stability analysis of the BDF slowest first multirate methods, CASAReport No 0704, 895–923

[19] V. Kurtc, I. Anufriev, “Local stability conditions and calibrating procedure for new carfollowing models used in driving simulators”, Proceedings of the 10th Conference on Traffic and Granular Flow'13, 2015, 453–461