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

ABSTRACT

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 $2\times 2$ junctions.

KEYWORDS:

• Shock discontinuities
• user-equilibrium conditions
• composite links unit
• bottleneck effects
• static traffic assignment

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The study was supported by grants from the National Natural Science Foundation of China (Grant Nos. 72101185, 72021002, 11972121), and the Research Grants Council of the Hong Kong Special Administrative Region, China (Grant No. 17204919). The sixth author was also supported by Francis S Y Bong Professorship in Engineering.