Analyses of lattice hydrodynamic models in traffic flow

Abstract

The present thesis is solely concerned with the development and analyses of mathematical models to understand traffic flow dynamics. We adopt the lattice hydrodynamic approach to model traffic phenomena, in which space is discretized in the form of lattices. In this work, we investigate the phenomenon of traffic congestion under the consideration of different important aspects such as driver’s behavior, passing phenomenon, interruption, gradient highways and multi-phase transition on one-dimensional single lane highway etc. Furthermore, single lane traffic flow model is also extended for two-lane system by incorporating the lane-changing behavior and the effect of driver’s behavior has been analyzed. The thesis also incorporates traffic models for networks on square as well as triangular lattices. All the proposed models are examined theoretically through linear as well as nonlinear stability analysis using the reductive perturbation method. The Burger, KdV and mKdV equation are derived near the critical point from nonlinear stability analysis and the formation of traffic congestion in terms of kink-antikink soliton density waves is explained. The critical boundaries are calculated theoretically for which kink solution of mKdV equation exists. To verify the theoretical results, numerical simulation is carried out by using finite difference scheme. The effects of sensitivity as well as other important parameters are discussed thoroughly and the results are also compared with those reported in the literature for specific choices of parameters. It is expected that LH models proposed in this thesis provide a certain degree of improvement to the existing models. We conclude the overall contribution of proposed work in traffic flow theory and provide some ideas about future extensions.