Fixed point methods for computing within-day dynamic user equilibrium

We show that analysis of the within-day dynamic user equilibrium (DUE) problem is tremendously simplified by expressing dynamic user equilibrium as a differential variational inequality when dynamic network loading (DNL) is considered to be an embedded subproblem. The DNL problem is approximated as a system of ordinary differential equations (ODEs) which may be efficiently solved using traditional numerical methods. Computing an actual dynamic user equilibrium is shown to require solution of a continuous-time fixed-point problem. A numerical example based on the much studied Sioux Falls network is presented.