Narrow Results

The traveling wave solution of a unified higher-order traffic flow model is investigated with a discontinuous fundamental diagram under the Lagrange coordinate. The equilibrium velocity is a piecewise function which consists of two concave functions. The weak solution theory is applied to study the traveling wave solution of the model, in which a set of equations about the characteristic parameters are obtained. Through numerical simulation, the moving cluster solutions of the anisotropic and isotropic traffic flow models are reproduced, respectively. The numerical results agree with the analytical ones.

The qualitative theory of differential equations is applied to investigate the traveling wave solution to an “anisotropic” higher-order viscous traffic flow model under the Lagrange coordinate system. The types and stabilities of the equilibrium points are discussed in the phase plane. Through the numerical simulation, the overall distribution structures of trajectories are drawn to analyze the relation between the phase diagram and the selected conservative solution variables, and the influences of the parameters on the system are studied. The limit-circle, limit circle-spiral point, saddle-spiral point and saddle-nodal point solutions are obtained. These steady-state solutions provide good explanation for the phenomena of the oscillatory and homogeneous congestions in real-world traffic.

The traffic bottleneck effects are investigated on a ring road with up- and down-slopes to the semi-discrete model under the Lagrange coordinate. The kinetic wave theory is applied to discuss all types of stationary solutions for a small relaxation time. Moreover, the analytical formulas of the queuing lengths before bottlenecks and the critical densities of the saturated flux are derived. The stability of the stationary solution on every road section under different mass increments is discussed. The relation between the slope gradient and saturated flux platform is also obtained. The numerical simulation reproduces many complex nonlinear phenomena, such as the steady-state flow, oscillatory congested traffic and saturated flux platform. This study helps to explain the empirical features on inhomogeneous roads in the real-world traffic.