A Continuum Model for Cities Based on the Macroscopic Fundamental Diagram: a Semi-Lagrangian Solution Method

Abstract

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.

Multi-directional continuous traffic model for large-scale urban networks

“…The drawback of this model is that the traffic density may become unbounded (it is not based on a fundamental diagram). There exist also other works [40,41] proposing 2D multi-layer models with bounded densities. However, they do not include mixing between different direction layers, i.e., vehicles can not change their direction of motion.…”

Effect of physical distancing on the speed–density relation in pedestrian dynamics

“…Following this approach, the use of FD as a way to study the capacity of collective systems was widely adopted and it started to be used in pedestrian dynamics. Nowadays, the FD is one of the most important macroscopic observables used in this field [6][7][8][9][10][11] allowing to establish comparisons between different experiments and validating numerical models. Furthermore, the FD has been applied in different scenarios and geometries, such as waiting rooms, stairs, sidewalks, or corridors with uni and bidirectional pedestrian flow [12].…”

Unit Sliding Mode Control for Disturbed Crowd Dynamics System Based on Integral Barrier Lyapunov Function

“…Hänseler et al [10] developed a macroscopic model to describe the dynamic of congested, multidirectional and time-varying pedestrian flows, where the anisotropy was accurately considered in the macroscopic framework. A consistent continuum macroscopic fundamental diagram model was formulated and solved with a semi-Lagrangian scheme in [11], where Eulerian and Lagrangian representations were both used. The dynamic of two intersecting pedestrian flows was modelled in [12] by using nonlinear partial differential equation, and then was illustrated with macroscopic and microscopic simulations.…”