A two-dimensional macroscopic model of traffic flows based on KCFD-schemes

The aim of investigations is development of a new two-dimensional model and computer simulation methods for vehicular traffic flows. At present scientific research in this field is worldwide very actual due to the traffic situation closing to the critical point. During daylight hours traffic on city roads and freeways becomes strongly congested. The average vehicular speed is not much higher than a pedestrian speed and is far from free-flow velocities, jams can take place, therefore drivers strategies are similar: to choose a lane with the lower vehicular density and to move with a speed providing safe traffic. In these conditions and under the condition that distances to be considered are much more length than vehicular sizes the continuum approach can be used. In the macroscopic theory individual vehicles do not appear explicitly. The traffic is viewed as a compressible fluid-dynamical flow: the notions of the density as a quantity of vehicles per lane in a distance unit and of the flux at an arbitrary location at an arbitrary instant of time are introduced. The initial model of this type belongs to M.H. Lighthill and G.B. Witham [1]. Traditionally traffic flow models are one-dimensional and describe flows only along one lane. Influence of neighboring lanes can be taken into account as sources and sinks in right-hand sides of the equations. The present paper is devoted to a two-dimensional model allowing predictions for real geometry of multi-lane roads. The model is constructed by analogy with the original kinetically consistent finite-difference (KCFD) schemes [2]. Formerly these algorithms were applied for viscous compressible and incompressible gas modeling and draw a parallel with the Lattice Boltzmann schemes and the stabilization technique by E. Onate [3]. In the developed model besides the traffic flow velocity the notion of the lateral velocity is introduced. It equals the average velocity of the vehicular transition from one lane to another. Consequently there is the lateral flux connected with the traffic transition to neighboring lanes. The equations for the density and the flux contain the corresponding terms [4]. On the basis of the developed and carefully studied mathematical model, numerical methods and computational algorithms realistic computer simulations of traffic flows will be performed involving the appropriate statistical information and empirical data about real objects

Incompressible viscous flow simulation using the double potential method

Abstract: In this paper we discuss the application of the double potential method for modeling an internal incompressible fluid flow. The resulting system of equations is approximated by using the finite volume method over cell centers and the exponential transformation. As a verification, the problem of establishing the Poiseuille flow on a flat tube and on a round one was used.Note: Research direction:Mathematical modelling in actual problems of science and technic