# Numerical Research of the Mathematical Model for Traffic Flow

A.S. KonkinaThe problems of distribution of transport flows are currently relevant in connection with the increase in vehicles. In the 50s of the last century, the first macroscopic (hydrodynamic) models appeared, where the transport flow resembles the flow "motivated" compressible liquid. The scientific approach based on the Navier-Stokes system. The main idea of the scholars is consideration the hydrodynamic models on the grounds of interrelation between the transport flow and incompressible fluid. For modelling traffic flows we examine Oskolkov equation on the geometric graph, where the edge has two positive values corresponding to it \textquotedblleft length\textquotedblright and \textquotedblleft width\textquotedblright. Certainly, in the context of mathematical model the values lk and bk are dimensionless, but for clarity it is convenient to imagine that lk is measured in linear metric units, for example, kilometers or miles, and bk is equal to the number of traffic lanes on the roadway in one direction. In terms of the Oskolkov model, we obtained a non-classical multipoint initial-final value condition. We will study such a model using the idea and methods of the Sobolean equation theory. These notes describe a numerical experiment based on the Galerkin method for the Oskolkov equation with a multipoint initial-final condition on the graph.Full text

- Keywords
- Oskolkov equation; geometric graph; multipoint initial-final condition; traffic flows.
- References
- 1. Hwang F.K., Richards D., Winter P. The Steiner Tree Problem. Amsterdam, Elsevier Science Publishers, 1992.

2. Siebel F., Mauser W. On the Fundamental Diagram of Traffic Flow. SIAM Journal on Applied Mathematics, 2006, no. 66, pp. 1150-1162.

3. Kurzhanski A.A., Kurzhanski A.B. Smart City Crossroads. Computer Research and Modeling, 2015, vol. 10, no. 3, pp. 347-358. DOI:10.20537/2076-7633-2018-10-3-347-358 (in Russian)

4. Zgonnikov A., Lubashevsky I. Double-Well Dynamics of Noise-Driven Control Activation in Human Intermittent Control: The Case of Stick Balancing. Cognitive Processing, 2015, vol. 16, no. 4, pp. 351-358. DOI: 10.1007/s10339-015-0653-5

5. Gorodokin V., Almetova Z., Shepelev V. Algorithm of Signalized Crossroads Passage Within the Range of Permissive-to-Restrictive Signals Exchange. Transportation Research Procedia, 2018, vol. 36, pp. 225-230. DOI: 10.1016/j.trpro.2017.01.05910.1007/s10339-015-0653-5

6. Banasiak J., Falkiewicz A., Namayanya P. Asymptotic State Lumping in Transport and Diffusion Problems on Networks with Applications to Population Problems. Mathematical Models and Methods in Applied Sciences, 2016, vol. 26, no. 2, pp. 215-247. DOI: 10.1142/S0218202516400017

7. Oskolkov A. P. Nonlocal Problems for Some Class Nonlinear Operator Equations Arising in the Theory Sobolev Type Equations. Zapiski LOMI, 1991, vol. 198, pp. 31-48. (in Russian)

8. Sviridyuk G.A., Zagrebina S.A., Konkina A.S. The Oskolkov Equations on the Geometric Graphs as a Mathematical Model of the Traffic Flow. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 148-154. DOI: 10.14529/mmp1503010

9. Oskolkov A.P. Nonlocal Problems for Some Class Nonlinear Operator Equations Arising in the Theory Sobolev Type Equations. Zapiski LOMI, 1991, vol. 198, pp. 31-48. (in Russian)

10. Zagrebina S.A., Konkina A.S. The Multipoint Initial-Final Value Condition for the Navier-Stokes Linear Model Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 1, pp. 132-136. DOI: 10.14529/mmp150111

11. Bayazitova A.A. On the Generalized Boundary-Value Problem for Linear Sobolev Type Equations on the Geometric Graph. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2018, vol. 10, no. 3, pp. 5-11.