Dynamic user equilibrium based on a hydrodynamic model

Terry L. Friesz, Ke Han, Pedro A. Neto, Amir Meimand, Tao Yao

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78 Scopus citations


In this paper we present a continuous-time network loading procedure based on the Lighthill-Whitham-Richards model proposed by Lighthill and Whitham (1955) and Richards (1956). A system of differential algebraic equations (DAEs) is proposed for describing traffic flow propagation, travel delay and route choices. We employ a novel numerical apparatus to reformulate the scalar conservation law as a flow-based partial differential equation (PDE), which is then solved semi-analytically with the Lax-Hopf formula. This approach allows for an efficient computational scheme for large-scale networks. We embed this network loading procedure into the dynamic user equilibrium (DUE) model proposed by Friesz et al. (1993). The DUE model is solved as a differential variational inequality (DVI) using a fixed-point algorithm. Several numerical examples of DUE on networks of varying sizes are presented, including the Sioux Falls network with a significant number of paths and origin-destination pairs (OD).The DUE model presented in this article can be formulated as a variational inequality (VI) as reported in Friesz et al. (1993). We will present the Kuhn-Tucker (KT) conditions for that VI, which is a linear system for any given feasible solution, and use them to check whether a DUE solution has been attained. In order to solve for the KT multiplier we present a decomposition of the linear system that allows efficient computation of the dual variables. The numerical solutions of DUE obtained from fixed-point iterations will be tested against the KT conditions and validated as legitimate solutions.

Original languageEnglish (US)
Pages (from-to)102-126
Number of pages25
JournalTransportation Research Part B: Methodological
StatePublished - Jan 2013

All Science Journal Classification (ASJC) codes

  • Civil and Structural Engineering
  • Transportation


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