Steady-State Solution And Bottleneck Efect In Trafc Flow Models Under The Lagrange Coordinate System

In this thesis, the weak solution theory, the qualitative theory of diferential equa-tion and kinetic wave theory are applied to study the anisotropic trafc flow modelsunder the Lagrange coordinate. An intrinsic link between the macroscopic and micro-scopic models is established through a semi-discrete model, and the steady-state solutionproperties of the viscous models are investigated. Moreover, the bottleneck efects is an-alyzed under inhomogeneous road conditions and trafc flow phenomena such as thewide moving jam, oscillatory congestion and narrow cluster are numerically simulated.The simulation also displays stationary flow, oscillatory congested and saturated fluxplatform in trafc. The contents of the thesis are briefed in the following.I. It proposes a semi-discrete model by using the concept of the Lagrange coordi-nate. The asymptotic relationship between the semi-discrete model and a macro-scopic higher-order model is indicated, which implies an intrinsic link between themacroscopic and microscopic models.We derive a semi-discrete model from the concept of the Lagrange coordinate. Thismodel could converge to a continuum higher-order model for the increment Mâ†'0,where M is the mass between two adjacent particles. And the linear stability regionof the former model could also converge to the later one. These results imply the con-sistency between the semi-discrete and continuum models. We numerically demonstratethat the semi-discrete model is able to reproduce a regular wide moving jam, and that forthe refinement of M its characteristic parameters do approach those that are analyticalderived by using the weak theory to the continuum model. The semi-discrete model re-duces to a car-following model for the increment M=1. Therefore, the semi-discretemodel could be a bridge between the macroscopic higher-order and microscopic car-following models, and the theories for these two types of models could be developed inparallel.II. The qualitative theory of diferential equation is applied to analyze the globalstructures of trajectories and the types of steady-state solutions in the phase planewith the viscous anisotropic trafc flow models under the Lagrange coordinate. Thenumerical solutions agree with the analytic ones.For the traveling wave solutions of the viscous anisotropic models, the qualitativetheory is applied to discuss the equilibrium points and their stabilities, and to analyzethe types of the steady-state solutions about the model. It is also remarkable that thestability of these equilibrium solution are discussed in two directions, i.e., with the inde-pendent variable extending to both positive and negative infinity. Through the numericalsimulation, we investigate the relation between the phase diagram and the selected con-servative solution variables, and discuss the influence about the parameters of the system.We also derive several global distributions of the trajectories and steady-state solutions which include the limit cycle, limit-spiral, saddle-limit cycle and saddle-spiral solutions.These solutions provide good explanations to the phenomena about the oscillatory andhomogeneous congestions in the real trafc. The numerical results of the correspondingmicroscopic car-following models display the steady-state solutions of the wide movingjam and narrow cluster solutions which agree with the results of the qualitative analysis.III. The stationary solution of the semi-discrete model is discussed on the highwaywith upgrade and downgrade sections. Simultaneously, the instable mechanismof the semi-discrete model is considered to analyze the stability of the stationarysolution.The bottleneck efects of the semi-discrete model are investigated on the highwaywith upgrade and downgrade sections. The influence of slopes is reflected in the relationof equilibrium velocity-density, which makes the equilibrium velocity turn into a space-dependent flux. Assuming the relaxation time is small, the stationary solution of thesemi-discrete model approaches to that of the LWR model. The kinetic wave theory isapplied to discuss all types of stationary solutions, and a set of algebraic equations of thecharacteristic parameters are obtained to determine the stationary solution. The analyticexpressions about the threshold of mass which can classify these stationary solutions, thelength of the queue before bottleneck and the critical density of the saturated flux are de-rived. The relation between the slope grade and saturated flux platform is also discussed.We consider the instability caused by the higher-order efects in the model to analyze thestability of the stationary solution on every road section. The numerical results repro-duce the stationary flow, oscillatory congested trafc and saturated flux platform, whichindicate the consistency with the analytical ones.