Unifiable multi-commodity kinematic wave model

:In the literature, many kinematic wave models have been proposed for multi-class vehicles on multi-lane roads; however, there lacks an explicit model of unifiable multi-commodity traffic, in which different commodity flows can have different speeds and violate the first-in-first-out (FIFO) principle, but there exists a speed-density relation for the total traffic. In this study, we attempt to fill the gap by constructing and solving a unifiable multi-commodity kinematic wave model. We first construct commodity speed-density relations based on generic generating functions. Then for two commodities we discuss the properties of the unifiable kinematic wave model and analytically solve the Riemann problem with a combination of total and commodity kinematic waves. We propose a unifiable multi-commodity Cell Transmission Model (CTM) with a general junction model for numerical simulations of network traffic flows, which are unifiable but may violate the FIFO principle. We prove that the CTM is well-defined under an extended CFL (Courant et al., 1928) condition. With examples we verify the consistency between the analytical and numerical solutions and demonstrate the convergence of the CTM. We conclude with several follow-up research directions for unifiable multi-commodity kinematic wave models.