Narrow Results

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.

The partial homogenization is a new method for the treatment of the boundary layers in the homogenization theory. It keeps the initial formulation near the boundary, passes to the high order homogenization at some distance from the boundary and prescribes the asymptotically precise junction conditions between the homogenized and the heterogeneous models at the interface. This method is related to the method of asymptotic partial domain decomposition MAPDD (see G. Panasenko, Method of asymptotic partial decomposition of domain, Math. Mod. Meth. Appl. Sci.8 (1998) 139–156). The partial homogenization (as well as the MAPDD) can be interpreted as a multi-scale model coupling the homogenized (macroscopic) description in the internal main part of the domain and the microscopic zoom in the domain of the location of the boundary layers. The semi-discretized partial homogenization uses some high order finite element projection in the homogenized subdomain.