A continuum model for cities based on the macroscopic fundamental diagram: A semi-Lagrangian solution method. (February 2020)

• Record Type: Journal Article
• Authors: Aghamohammadi, Rafegh; Laval, Jorge A.
• Is Part Of: Transportation research. Volume 132(2020)
• Page Start: 101
• Page End: 116
• Publication Date: 2020-02
• Subjects: Continuum modeling -- Macroscopic fundamental diagram -- Semi-Lagrangian -- Anisotropic; Transportation -- Research -- Periodicals; Transportation -- Mathematical models -- Periodicals
• DOI: 10.1016/j.trb.2019.04.011
• Languages: English
• ISSNs: 0191-2615

Abstract

Highlights: The connection between continuum-space DTA models and MFD is not well-understood. This paper proposes a consistent continuum model with MFD theory. The consistent model incorporates randomly-distributed destination points over the continuum space. The proposed semi-Lagrangian solution method is capable of handling anisotropy. Two numerical examples show the convergence properties of the 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.