Two-dimensional macroscopic model of traffic flows

Sukhinova, A. B.; Trapeznikova, Marina A.; Четверушкин, Б. Н.; Churbanova, N. G.

doi:10.1134/s2070048209060027

Cited by 18 publications

(18 citation statements)

References 14 publications

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“…This allows us to simulate the actual road transport situation with a greater freedom and better justification in its mathematical description. For example, the state equation p = const · ρ 2 often used for so-called 'transport pressure' in some hydrodynamic transport models (see, e.g., [6,14,32]) follows from equation (3.15) p q = const · ρ 1+2/D q for the dimension D = 2 of the phase space of velocities.…”

Section: Resultsmentioning

confidence: 99%

“…The so-called 'quasi-hydrodynamic model' of vehicle traffic in the domain of synchronized flows [6,17,32] proves efficient for the traffic flow description for streets of large towns and multiline highways (when all drivers are forced to follow the same strategy of behaviour and obey the same rules). This model is intermediate (mesoscale) between pure kinetic and hydrodynamic ones and can be obtained on the base of a quasi-hydrodynamic (QHD) system of equations [7,8,13,31] originating, in its turn, from the modernized Bhatnagar-Gross-Krook (BGK) kinetic equation.…”

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confidence: 99%

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Kinetic derivation of a quasi-hydrodynamic system of equations on the base of nonextensive statistics

Колесниченко

Четверушкин

2013

Russian Journal of Numerical Analysis and Mathematical Modelling

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Applying the formalism of the Tsallis nonadditive statistical mechanics, a derivation of hydrodynamic and quasi-hydrodynamic equations is considered based on the generalized BGK kinetic equation. In particular, those equations are intended for construction of more appropriate macroscopic and mesoscopic models of transport systems related to the so-called anomalous systems where the corresponding phase space possesses a complicated (fractal) structure. The classic Boltzmann-Gibbs statistics (or Maxwell's distribution of velocities in the case of kinetic theory) is violated in such systems and hence the additivity of extensive variables related to it does not hold either. The application of the approach developed in this paper does not change the structure of hydrodynamic and quasihydrodynamic equations, but the modified thermal and calorific state equations and also the transfer coefficients contain two additional free parameters, which are the parameter q of the non-additivity of the system and the fractional dimension D of the phase space. Along with the smoothing parameter τ, these parameters can be determined empirically in each particular case from statistical or experimental data, which allows us to simulate the actual variable traffic situation within a continual approach both in regular and crisis cases.By the analogy with gas dynamics, I. Prigogine (in cooperation with F. Andrews and R. Herman) proposed half a century ago to describe traffic flows (associated in this case with a compressible fluid with motivation) by the generalized kinetic Boltzmann's equation where, however, an 'integral of vehicle interaction' is used instead of the integral of gas particle collision [25][26][27][28][29]. Macroscopic (hydrodynamic) equations for traffic flows can be obtained from Prigogine's kinetic equation for the distribution function of vehicles similar to the derivation of the Navier-Stokes equation in the kinetic gas theory, i.e., by its averaging with additive invariants over the phase space. This approach was essentially refined and improved in the papers of Paveri-Fontana [23], Helbing [15], and some other authors *

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Section: Resultsmentioning

confidence: 99%

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confidence: 99%

Kinetic derivation of a quasi-hydrodynamic system of equations on the base of nonextensive statistics

Колесниченко

Четверушкин

2013

Russian Journal of Numerical Analysis and Mathematical Modelling

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“…3. 5 show calculation results obtained using the macroscopic model [18]. The density of cars decreases on the wider part of the road (Fig.…”

Section: Simulating Two Dimensional Traffic Flowmentioning

confidence: 98%

“…To check whether the proposed model is adequate, we performed test calculations using the micro scopic and macroscopic methods [17,18]. of propagation of the density jump along the road obtained based on macro and microscopic models, respectively (here, and in what follows, the length of the road is given in km for the macroscopic model and in m for the microscopic model).…”

Section: Simulating One Dimensional Traffic Flowmentioning

confidence: 99%

“…In [17,18], a two dimensional multilane macroscopic model of traffic flows was proposed that describes the real geometry of the road and uses ideology that is the basis of quasi gas dynamic equations [19]. In this work, we propose a microscopic approach that is also generalized for multilane motion and based on cellular automata theory [11].…”

Section: Introductionmentioning

confidence: 99%

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