Modeling space–time inhomogeneities with the kinematic wave theory

In this paper, we are concerned with modeling space–time inhomogeneities with the kinematic wave (LWR) model. The notion of space–time inhomogeneity refers to the fact that governing laws of traffic, essentially dictated by fundamental diagrams (FD), differ from each other in distinct space–time regions. Such a scenario is common when exogenous inputs, e.g. a group of slowly moving vehicles, emerge in the modeling process. We will prove the well-posedness of this class of problems. More importantly, we show that if the boundary delineating two neighboring regions is continuous and has bounded speed, this problem can be greatly simplified by introducing a piecewise linear approximation to the boundary. In particular, we utilize the variational formulation of the kinematic wave model and prove that this approximation results in uniformly bounded errors in cumulative flow N which are proportional to the L∞ deviation of the approximation. The numerical solution of this simplified problem is well understood, and this result means that a kinematic wave model with space–time inhomogeneity can be solved accurately with any existing Godunov type scheme. Finally, using the inhomogeneous LWR model, we explain the capacity drop as a natural result of space–time inhomogeneity.