A shock-fitting algorithm for the Lighthill–Whitham–Richards model on inhomogeneous highways

Abstract

The analytical shock-fitting algorithm outperforms traditional numerical methods in solving the Lighthill–Whitham–Richards (LWR) traffic flow model for homogeneous highways. In this study, we extend the algorithm to an inhomogeneous highway in which two homogeneous sections with different fundamental diagrams are connected by a junction (interface). According to the entropy condition of the Riemann problem, the flow conditions at the interface can be categorized into four groups, and the density on both sides of the interface can be uniquely determined. Based on the physical conditions, we construct a fictitious element as an appropriate boundary condition for each homogeneous section. Consequently, the Riemann problem of an inhomogeneous highway can be transformed into a problem of equivalent homogeneous sections to which the shock-fitting algorithm can be applied. We apply this algorithm to some representative traffic flow cases and compare the results with numerical solutions obtained using the Weighted Essentially Non-Oscillatory (WENO) scheme.