In this paper we present a continuous-time network loading procedure based on the Lighthill–Whitham–Richards model proposed by Lighthill and Whitham, 1955, 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.