Nonlinear hydrodynamic models of traffic flow modelling and mathematical problems

This paper deals with nonlinear hydrodynamic modelling of traffic flow on roads and with the solution of related nonlinear initial and boundary value problems. The paper is in two parts. The first one provides the general framework of hydrodynamic modelling of traffic flow. Some new models are proposed and related to the ones which are known in the literature. The second one is on mathematical methods related to the solution of initial-boundary value problems. A critical analysis and an overview on research perspectives conclude the paper.     

New mathematical models of porous flow

A new, viable mathematical approach to the analysis of porous flow is developed. Liquids and solids are modelled as sets of particles which interact in accordance with both long range (gravity) and short range (molecular type) forces. The resulting nonlinear, n-body problem is resolved numerically on a digital computer. A variety of fundamental porous flow type problems, solved on the University of Texas at Arlington IBM 370-155, are described and analysed.     

Chattering in dynamic mathematical two-phase flow models

This paper presents a chattering problem which arises in a dynamic mathematical two-phase flow model. The real system under study is also introduced, the DISS test facility, a parabolic-trough solar thermal power plant. The heat transfer fluid in the DISS facility is the steam-water mixture. A dynamic model of this plant, using Modelica as the modeling language, was previously developed in order to study its behavior. Chattering arises in the pipe model reducing the computational performance and hence limiting the applicability of the model. The problem source is studied and analysed together with an approach to the problem which is based on the smooth interpolation of some thermodynamic properties.     

Global well posedness of traffic flow models with phase transitions

This note is devoted to the study of traffic flow models that develop phase transitions. From the analytical point of view, this is a first example of a well posedness result for conservation laws developing phase transitions, which is independent from the number of phase boundaries in the initial data or in the solutions. We consider below the Cauchy problem as well as the problem with boundaries.     

Nonlinear hydrodynamic models of traffic flow in the presence of tollgates

A nonlinear hydrodynamic model of traffic flow is here proposed in order to refine the modeling of driver's behaviour. This model is able to describe the car density and flow evolution in the presence of tollgates too. In any case, the associate evolution problem is a classical Dirichlet problem related to the flow measurement. Numerical simulations related to the solution of initial-boundary value problems are obtained by employing a scientific program written by the software Mathematica.     

Hydrodynamic models of traffic flow: Drivers' behaviour and nonlinear diffusion

This paper deals with scalar hydrodynamic models of traffic flow on roads. The equivalence between models with diffusion and models with a driver is analyzed, putting in evidence equivalences and contradictions. A new model is proposed in order to refine the modelling of the driver's behaviour.     

Differential transform method for mathematical modeling of jamming transition problem in traffic congestion flow

In this paper we aim to find an analytical solution for jamming transition in traffic flow. Generally the Jamming Transition Problem (JTP) can be modeled via Lorentz system. So, in this way, the governing differential equation achieved is modeled in the form of a nonlinear damped oscillator. In current research the authors utilized the Differential Transformation Method (DTM) for solving the nonlinear problem and compared the analytical results with those ones obtained by the 4th order Runge-Kutta Method (RK4) as a numerical method. Further illustration embedded in this paper shows the ability of DTM in solving nonlinear problems when a so accurate solution is required.     

On the velocity distribution at equilibrium in the mathematical theory of vehicular traffic flow

This work deals with a two-scale vehicular traffic model, where the mass conservation equation is closed by a velocity probability distribution, parameterized with respect to each density of vehicles. The above probability distribution is identified on the basis of both experimental information and theoretical conjectures.     

Thresholds for shock formation in traffic flow models with nonlocal-concave-convex flux

We identify sub-thresholds for finite time shock formation in a class of non-local conservation law with concavity changing flux. From a class of non-local conservation laws, the Riccati-type ODE system that governs a solution's gradient is obtained. The changes in concavity of the flux function correspond to the sign changes in the leading coefficient functions of the ODE system. We identify the blow up condition of this structurally generalized Riccati-type ODE. The method is illustrated via the traffic flow models with nonlocal-concave-convex flux. The techniques and ideas developed in this paper is applicable to a large class of non-local conservation laws.     

The mathematical solution of a cellular automaton model which simulates traffic flow with a slow-to-start effect

In this paper we investigate a cellular automaton model associated with traffic flow and of which the mathematical solution is unknown before. We classify all kinds of stationary states and show that every state finally evolves to a stationary state. The obtained flow-density relation shows multiple branches corresponding to the stationary states in congested phases, which are essentially due to the slow-to-start effect introduced into this model. The stability of these states is formulated by a series of lemmas, and an algorithm is given to calculate the stationary state that the current state finally evolves to. This algorithm has a computational requirement in proportion to the number of cars.     

Parametric scaling of mathematical models

Parametric scaling, the process of extrapolation of a modelling result to new parametric conditions, is often required in model optimization, and can be important if the effects of parametric uncertainty on model predictions are to be quantified. Knowledge of the functional relationship between the model solution (y) and the system parameters (α) may also provide insight into the physical system underlying the model. This paper examines strategies for parametric scaling, assuming that only the nominal model solution y(α) and the associated parametric sensitivity coefficients (?y/?α, ?²y/?α², etc.) are known. The truncated Taylor series is shown to be a poor choice for parametric scaling, when y has known bounds. Alternate formulae are proposed which ‘build-in’ the constraints on y, thus expanding the parametric region in which the extrapolation may be valid. In the case where y has a temporal as well as a parametric dependence, the extrapolation may be further improved by removing from the Taylor series coefficients the ‘secular’ components, which refer to changes in the time scale of y(t), not to changes in y as a function of α.     

Some mathematical models related to the Zhukovskii problem of the flow around a sheet pile

The seepage under a Zhukovskii sheet pile through a layer of soil underlain by a highly permeable pressurized horizon is considered. The left semi-infinite part of the roof of this horizon is simulated by an impermeable foundation. The flow when the velocity on the edges of the sheet pile is equal to infinity and, on the two water permeable parts of the boundary of the domain of motion, the flow rate takes extremal values, is investigated. The limiting cases, associated with the absence of both a backwater and an impermeable inclusion, are mentioned. The problem of seepage from a foundation pit formed by two Zhukovskii sheet piles is solved within the limits of a flow with a highly permeable pressurized stratum lying below. In the case when there is no infiltration onto the free surface, a solution of the well-known Vedernikov problem is obtained. A contact scheme, arising when there are no such indicated critical points, is considered; it is described outside the scope of the constraints imposed on the unknown conforming mapping parameters ensuring the realization of the basic mathematical model. Solutions are given for two schemes of motion in a semi-inverse formulation. The classical Zhukovskii problem is the limiting case of one of them. The special features of such models are mentioned. The Polubarinova-Kochina method is used to study all the above-mentioned flows. This method enables exact analytical representations of the elements of the motion to be obtained. The results of numerical calculations and an analysis of the effect of all the physical factors on the seepage characteristics are presented.     

Fundamental mathematical structures of integrable models

We consider integrable models in a totally discrete multidimensional space-time. Dynamic variables are associated with cells into which the space is decomposed by a set of intersecting hyperplanes. We investigate the (2+1)-dimensional model related to the functional tetrahedron equation. We propose a method for constructing solutions of analogous models in higher dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 405–412, March, 1999.     

Non-linear analysis of a traffic flow

A nonlinear mathematical model, which takes into account the dissipative mechanism, is used to describe the signal transmission in a traffic flow. It is shown that dissipative mechanisms, under certain conditions, may produce attenuation effects against the typical nonlinear steepening of waves. An asymptotic analysis is carried out to discuss wave features when the governing hyperbolic system of equations is objective to different kinds of approximations.     

Characterizing optimality in mathematical programming models

This paper is a survey of basic results that characterize optimality in single- and multi-objective mathematical programming models. Many people believe, or want to believe, that the underlying behavioural structure of management, economic, and many other systems, generates basically continuous processes. This belief motivates our definition and study of optimality, termed structural optimality. Roughly speaking, we say that a feasible point of a mathematical programming model is structurally optimal if every improvement of the optimal value function, with respect to parameters, results in discontinuity of the corresponding feasible set of decision variables. This definition appears to be more suitable for many applications and it is also more general than the usual one: every optimum is a structural optimum but not necessarily vice versa. By characterizing structural optima, we obtain some new, and recover the familiar, optimality conditions in nonlinear programming.The paper is self-contained. Our approach is geometric and inductive: we develop intiution by studying finite-dimensional models before moving on to abstract situations.Research partly supported by the National Research Council of Canada.     

Optimal control laws for traffic flow

Optimal “on–off” laws for the traffic signals are developed based on the bilinear control problem with the binary constraints. A Lyapunov function based feedback law for regulating traffic congestions is developed. Also, a real-time optimal signal law is developed using a novel binary optimization method. Both methods are tested and compared, and our tests demonstrate that the both methods provide very effective and efficient traffic control laws.     

Modeling traffic flow interrupted by incidents

A steady-state M/M/c queueing system under batch service interruptions is introduced to model the traffic flow on a roadway link subject to incidents. When a traffic incident happens, either all lanes or part of a lane is closed to the traffic. As such, we model these interruptions either as complete service disruptions where none of the servers work or partial failures where servers work at a reduced service rate. We analyze this system in steady-state and present a scheme to obtain the stationary number of vehicles on a link. For those links with large c values, the closed-form solution of M/M/∞ queues under batch service interruptions can be used as an approximation. We present simulation results that show the validity of the queueing models in the computation of average travel times.