  This paper presents a formulation of the reactive dynamic user equilibrium
problem in continuum form using a network-level Macroscopic Fundamental Diagram
(MFD). Compared to existing continuum models for cities -- all based in Hughes'
pedestrian model in 2002 -- the proposed formulation (i) is consistent with
reservoir-type models of the MFD literature, shedding some light into the
connection between these two modeling approaches, (ii) can have destinations
continuously distributed on the region, and (iii) can incorporate
multi-commodity flows without additional numerical error. The proposed
multi-reservoir numerical solution method treats the multi-commodity component
of the model in Lagrangian coordinates, which is the natural representation to
propagate origin-destination information (and any vehicle-specific
characteristic) through the traffic stream. Fluxes between reservoir boundaries
are computed in the Eulerian representation, and are used to calculate the
speed of vehicles crossing the boundary. Simple examples are included that show
the convergence of the model and its agreements with the available analytical
solutions. We find that (i) when origins and destinations are uniformly
distributed in a region, the distribution of the travel times can be
approximated analytically, (ii) the magnitude of the detours from the optimal
free-flow route due to congestion increase linearly with the inflow and
decreases with the square of the speed, and (iii) the total delay of vehicles
in the network converges to the analytical approximation when the size of
reservoirs tends to zero.
