Stationary solution of semi-discrete traffic flow model on a ring road with slopes

Abstract

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.