Steady-state solution to the LWR model on a single origin-destination parallel road network

This paper describes the systematical study of a steady-state solution to the Lighthill-Witham-Richards (LWR) model on a single origin-destination parallel road network, in which the user-equilibrium condition is satisfied and shocks on links are permitted. This study derives a novel static traffic assignment model considering the complete fundamental diagram, including the congested branch. For a composite links unit that includes a set of parallel links between two junctions, the user-equilibrium condition is discussed in detail and thus, the so-called user-equilibrium curves are defined. For a single origin-destination parallel road network, we note that shock structures must be introduced to guarantee the existence of the solution when a bottleneck exists, and thus we establish the correlation between the total number of vehicles and the steady-state solution. Moreover, the uniqueness of the solution is proved by introducing priority coefficients when shocks appear. We analytically give the complete solving procedure of the steady-state solution and thus, avoid the iterative algorithms used in other static traffic assignment models. A numerical scheme of the LWR network model is designed to converge the traffic flow into the discussed steady-state solution, by determining the percentages and priority coefficients at junctions. A numerical example is given to validate the convergence for the designed numerical scheme on a road network with two $ 2\times 2 $ 2x2 junctions.