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

## TL;DR

This paper introduces a continuum city traffic model based on the Macroscopic Fundamental Diagram, offering a novel numerical solution method that handles multi-commodity flows and continuous destinations, improving accuracy and analytical insight.

## Contribution

It presents a new continuum formulation for city traffic modeling that integrates reservoir-type models with multi-commodity flows using a semi-Lagrangian numerical method.

## Key findings

- Model convergence demonstrated with simple examples.
- Travel time distribution approximated analytically under uniform distributions.
- Congestion-induced detours increase linearly with inflow and decrease with speed.

## 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.

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Source: https://tomesphere.com/paper/1902.05057