This paper proposes two lattice traffic models by taking into account the drivers＇ delay in response. The lattice versions of the hydrodynamic model are described by the differential-difference equation and difference-difference equation, respectively. The stability conditions for the two models are obtained by using the linear stability theory. The modified KdV equation near the critical point is derived to describe the traffic jam by using the reductive perturbation method, and the kink-antikink soliton solutions related to the traffic density waves are obtained. The results show that the drivers＇ delay in sensing headway plays an important role in jamming transition.
The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.
Car following model is one of microscopic models for describing traffic flow. Through linear stability analysis, the neutral stability lines and the critical points are obtained for the different types of car following models and two modified models. The singular perturbation method has been used to derive various nonlinear wave equations, such as the Kortewegde-Vries （KdV） equation and the modified Korteweg-de-Vries （mKdV） equation, which could describe different density waves occurring in traffic flows under certain conditions. These density waves are mainly employed to depict the formation of traffic jams in the congested traffic flow. The general soliton solutions are given for the different types of car following models, and the results have been used to the modified models efficiently.
