A new traffic kinetic model for heterogeneous condition Shoufeng Lu n , Gaihong Liu, Ximin Liu, Wei Shao Traffic and Transportation Engineering College, Changsha University of Science and Technology, Changsha, Hunan, China article info Article history: Received 4 December 2012 Received in revised form 31 March 2013 Accepted 1 April 2013 Available online 16 April 2013 Keywords: Vehicular traffic flow Kinetic Theory of Active Particles Spatially heterogeneous Cell transmission model abstract The paper aims to integrate Cell Transmission Model (CTM) and the Delitala–Tosin model of a homogeneous condition based on the so-called Kinetic Theory of Active Particles (KTAP) to model the heterogeneous condition. The integrations overcome solution of partial differential equations, and transforms to solution of ordinary differential equations. The deficiency of solving partial differential equations is that an improper difference scheme can cause instability and non-convergence. In order to consider the difference in local densities, space variable is also discrete in the paper. In order to take the effect of distance on interaction into account, the paper introduces law of gravity to model interaction. Finally, we give some numerical result of four heterogeneous traffic cases and compare them with those treated in the paper by Delitala–Tosin where the fixed grid is used and by Coscia–Delitala–Frasca where the adaptive grid is used. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Methods of Kinetic Theory of Active Particles (KTAP) had been developed to model vehicular traffic flow. This approach on the one hand converts the Boltzmann's integral-differential equation into a set of partial differential equations, on the other hand relaxes the continuum hypothesis and includes the granular nature of vehicular traffic. Further details on methods of KTAP are explained in [1–3]. Kinetic-type models with discrete velocities for traffic flow have been proposed. There are three methods of discrete velocity. The first one uses fixed velocity grid [4]. The second one uses adaptive velocity grid [5]. The third one considers the coupling of fixed velocity grid and adaptive velocity grid [6]. For the third method, the number of velocity classes is constant, and when the density is less than critical density, velocity is discrete by a fixed grid. When the density is larger than critical density, velocity is discrete by an adaptive grid. Bonzani and Mussone [7] deal with identification of the parameters of Delitala–Tosin model using experimental data obtained on Padova–Venezia highway. Vehicular traffic flow is composed of many driver–vehicle units. The driver–vehicle units, which are called active particles, can modify their dynamics according to specific strategies due to their ability which are different from classical particles in Newtonian dynamics. Gramani [8], Bellouquid et al. [9] modeled driver–vehicle unit as such an active particle. In particular, these two papers include in the generalized velocity distribution function an activity variable, which describes the driving skills, to model the individual behaviors. In general, three types of vehicles are involved in the interac- tions: test vehicles which are representatives of the whole system, field vehicles which interact with test and candidate vehicles, and candidate vehicles which may acquire, with a certain probability, the state of a test vehicle by interaction with the field vehicles. The main tools of discrete mathematical kinetic theory are composed of encounter rate and table of games. Encounter rate and table of games are two terms appeared in references [1–9], here we follow this usage. Encounter rate describes the number of interactions per unit time with different velocity. For example, η hk denotes the encounter rate of vehicles with velocities V h and V k . Table of games describes the velocity transition probability after vehicle interac- tion. For example, A i hk is the probability that the candidate vehicle with velocity V h reaches the velocity V i , after the interaction with the field vehicle with velocity V k . The mathematical structure of evolution equations is that the variation of velocity distribution is equal to increase amount minus decrease amount. Because of the property of table of games Σ n i ¼ 1 A i hk ¼ 1; h; k ¼ 1; 2; ⋯n, the series of model of discrete mathematical kinetic theory are conserved. For the spatially heterogeneous case, evolution equation is a partial differential equation of hyperbolic conservation laws. The solution scheme of hyperbolic conservation laws with source terms can obtain a numerical solution, but the error increases with solution time step, more details refer to Toro [10]. Aim of the paper is to build a model integrating Cell Transmis- sion Model (CTM) with spatially homogeneous KTAP model to deal with spatially heterogeneous traffic flow conditions. At first we intend to integrate CTM with the model based on adaptive grids but we rapidly confess the impossibility to reach this goal. Then, Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics 0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.04.001 n Corresponding author. Tel.: +86 731 899 10812. E-mail address: itslusf@gmail.com (Lu Shoufeng). International Journal of Non-Linear Mechanics 55 (2013) 1–9