ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Comparative Analysis of Different Numerical Methods for the Quasilinear Equation of the Traffic Flow

### Author(s):

Postgraduate Student of Department of Mathematical Physics,
Faculty of Computational Mathematics and Cybernetics,
Lomonosov Moscow State University.
CMC Faculty, Lomonosov MSU, Leninskyie Gori, Moscow, Russia 119991.

anastasiapodoroga@gmail.com

Doctor of Physical and Mathematical Sciences,
Professor of Department of Mathematical Physics,
Faculty of Computational Mathematics and Cybernetics,
Lomonosov Moscow State University.
CMC Faculty, Lomonosov MSU, Leninskyie Gori,
Moscow, Russia 119991

ivtikh@mail.ru

### Abstract:

The paper presents an extended description of the report which was made at the scientific conference «Herzen Readings - 2017». Our study is devoted to the mathematical theory of the traffic flow. A macroscopic approach is used. We consider a quasilinear equation of the traffic flow and provide the necessary theoretical information. Four numerical methods for solving the equation are discussed: difference schemes, method of characteristics, method of particles and a new method of shocks movement which we propose. For each of these methods the main idea is stated, the typical features with advantages and disadvantages are noted. Also we provide illustrations to give examples of the specified computer methods. The numerical experiments allow us to formulate a special corollary about the stabilization of traffic flows on a ring road.

### Keywords

• mathematical theory of traffic flow
• numerical methods
• quasilinear differential equations
• stabilization of solutions

### References:

1. Uizem Dzh. Linejnye i nelinejnye volny. - M. : «Mir», 1977. - 624 s
2. Inosje H., Hamada T. Upravlenie dorozhnym dvizheniem. - M. : «Transport», 1983. - 248 s
3. Treiber M., Kesting A. Traffic flow dynamics. Data, models and simulation. - Berlin: Springer-Verlag, 2013. - 506 p
4. Gasnikov A. V. i dr. Vvedenie v matematicheskoe modelirovanie transportnyh potokov: Uchebnoe posobie / Pod red. A. V. Gasnikova. Izdanie 2-e, ispr. i dop. - M. : MCNMO, 2013. - 427 s
5. Laks P. D. Giperbolicheskie differencial'nye uravnenija v chastnyh proizvodnyh. - M. -Izhevsk: NIC «Reguljarnaja i haoticheskaja dinamika», 2010. - 296 s
6. Jevans L. K. Uravnenija s chastnymi proizvodnymi. - Novosibirsk: Tamara Rozhkovskaja, 2003. - 576 s
7. Gorickij A. Ju., Kruzhkov S. N., Chechkin G. A. Uravnenija s chastnymi proizvodnymi pervogo porjadka. (Uchebnoe posobie). - M. : Meh-mat MGU, 1999. - 96 s
8. Greenberg H. An analysis of traffic flow // Operations Research. 1959. Vol. 7. No. 1. - P. 79-85
9. Nagel K., Schreckenberg M. A cellular automaton model for freeway traffic // Journal de Physique I France. 1992. Vol. 2. No. 12. - P. 2221-2229
10. Podoroga A. V., Tihonov I. V. Kvazilinejnoe uravnenie dorozhnogo dvizhenija i komp'juternoe modelirovanie // Nekotorye aktual'nye problemy sovremennoj matematiki i matematicheskogo obrazovanija. Gercenovskie chtenija - 2015. - SPb. : izd-vo RGPU im. A. I. Gercena, 2015. - S. 209-213
11. Podoroga A. V., Tihonov I. V. O predel'nyh sostojanijah zamknutyh transportnyh potokov na kol'cevoj avtodoroge // Nekotorye aktual'nye problemy sovremennoj matematiki i matematicheskogo obrazovanija. Gercenovskie chtenija - 2016. - SPb. : izd-vo RGPU im. A. I. Gercena, 2016. - S. 222-228. (Sm. takzhe: Differencial'nye uravnenija i processy upravlenija. 2016. № 2. Materialy konferencij. - S. 271-279. )
12. Smirnov N. N., Kiselev A. B., Nikitin V. F, Jumashev M. V. Matematicheskoe modelirovanie avtotransportnyh potokov. - M. : Meh-mat MGU, 1999. - 31 s
13. Bahvalov N. S., Zhidkov N. P., Kobel'kov G. M. Chislennye metody / 7-e izd. - M. : BINOM. Laboratorija znanij, 2013. - 636 s
14. Lighthill M. J., Whitham G. B. On kinematic waves. II. A theory of traffic flow on long crowded roads // Proceedings of the Royal Society of London. Ser. A, Math. And Physical Sciences. 1955. Vol. 229. No. 1178. - P. 317-345
15. Bogomolov S. V., Zamaraeva A. A., Karabelli H., Kuznecov K. V. Konservativnyj metod chastic dlja kvazilinejnogo uravnenija perenosa // Zhurnal vychisl. matem. i matem. fiziki. 1998. T. 38. № 9. - S. 1602-1610
16. Baev A. Zh., Bogomolov S. V. Ob ustojchivosti razryvnogo metoda chastic dlja uravnenija perenosa // Matematicheskoe modelirovanie. 2017. T. 29. № 9. - S. 3-18
17. Podoroga A. V. Modificirovannyj metod chastic dlja kvazilinejnogo uravnenija dorozhnogo dvizhenija // Sistemy komp'juternoj matematiki i ih prilozhenija: materialy XVIII Mezhdunarodnoj nauchnoj konferencii, posvjashhennoj 70-letiju V. I. Munermana. - Smolensk: Izd-vo SmolGU, 2017. Vyp. 18. - S. 30-33
18. Olejnik O. A. O edinstvennosti i ustojchivosti obobshhennogo reshenija zadachi Koshi dlja kvazilinejnogo uravnenija // Uspehi matem. nauk. 1959. T. 14. № 2 (86). - S. 165-170
19. Keyfitz (Quinn) B. Solutions with shocks: an example of an contractive semigroup // Communications on Pure and Applied Mathematics. 1971. Vol. 24. - P. 125-132