A nonconservative macroscopic traffic flow model in a two-dimensional urban-porous city

TL;DR

The paper develops a novel nonconservative macroscopic traffic model for urban domains viewed as porous media, coupling density transport with Darcy–Brinkman–Forchheimer momentum dynamics to capture interchanges between streets and off-street parking. A modified Eikonal framework provides a density-dependent desired speed, implemented via a linearized potential $\psi$ and a velocity field $\mathbf{v}(\rho)$, while diffusion and viscosity terms stabilize the numerical scheme. The authors formulate a full variational finite element treatment using $P_1$ elements and an SSP explicit time-stepping method, enabling stable simulations on a Guadalajara-inspired domain with varying porosity. Numerical experiments illustrate how porosity, relaxation time $\tau$, parking absorption rate $\kappa$, and time-varying demand $g(t)$ influence traffic density and speed, showing faster flows in more porous (streets-dense) cities and significant center congestion under limited parking. The framework offers a first-principles, scalable tool for city-scale traffic planning and pollution assessment, with potential extensions to multiple attraction points and parking networks.

Abstract

In this paper we propose a novel traffic flow model based on understanding the city as a porous media, this is, streets and building-blocks characterizing the urban landscape are seen now as the fluid-phase and the solid-phase of a porous media, respectively. Moreover, based in the interchange of mass in the porous media models, we can model the interchange of cars between streets and off-street parking-spaces. Therefore, our model is not a standard conservation law, being formulated as the coupling of a non-stationary convection-diffusion-reaction PDE with a Darcy-Brinkman-Forchheimer PDE system. To solve this model, the classical Galerkin P1 finite element method combined with an explicit time marching scheme of strong stability-preserving type was enough to stabilize our numerical solutions. Numerical experiences on an urban-porous domain inspired by the city of Guadalajara (Mexico) allow us to simulate the influence of the porosity terms on the traffic speed, the traffic flow at rush-valley hours, and the streets congestions due to the lack of parking spaces.

Figures (9)

• Figure 1: (\ref{['fig1:GDL']}) A city as an urban-porous media domain: city landscape is characterized by building-blocks delimited by streets, in this case in Guadalajara city. (\ref{['fig1:diagram']}) Scheme of the porous traffic flow model: cars leave residential buildings (green arrows) on suburbia areas (in blue) to enter the city center (in red), flowing on the urban-porous media towards the attraction point represented by a yellow cross (black arrows). This attraction point belongs to a zone containing malls, working places, or schools, were cars leave the fluid phase going into the corresponding parking-spaces (magenta arrows).
• Figure 2: (\ref{['fig2:ZMGDL']}) The numerical domain is inspired by the metropolitan zone of Guadalajara (Mexico). The satellite image of the zone under study shows the urban limit in yellow, and the different obstacles in green, red, purple, and orange (Google Earth, 2024). (\ref{['fig2:mesh']}) Triangular mesh $\mathcal{K}_h$ used for the domain, depicting the urban limit boundary $\Gamma_{bnd}$, and the obstacle walls $\Gamma_w$.
• Figure 3: Inputs and initial state of the traffic density for our model: (\ref{['fig3:G']}) Normalized function $G$ used to enforce Eikonal potential $\phi$ to target the city center. (\ref{['fig3:initialdensity']}) Initial state of the traffic density $\rho^0$, constructed under the idea of a concentric city. (\ref{['fig3:eikonal']}) Eikonal potential $\phi$ derived from the definition of $G$. (\ref{['fig3:desiredspeedvector']}) Detail of the vector field for the desired speed $\mathbf{v}(\rho^0)$, pointing to city center and avoiding obstacles. (\ref{['fig3:desiredspeedvelocity']}) Euclidean norm of $\mathbf{v}(\rho^0)$, reaching -but not surpassing- maximum velocity $U_{max}=50$. (\ref{['fig3:kappa']}) Absorption rate distribution in the whole city, assuming -as in previous parameters- a concentric structure of the city.
• Figure 4: The two city scenarios are defined by their urban porosity values: (\ref{['fig4:CiudadDensa']}) The dense building city presents a porosity value at center $\epsilon_c = 0.38$. (\ref{['fig4:CiudadDispersa']}) The disperse building city shows a porosity value at the city center $\epsilon_c=0.62$. In both cases, the city environs have a porosity maximum value of $\epsilon_{max} = 0.82$.
• Figure 5: Traffic density and speed for the two city scenarios at the same simulation time $t=0.25$ h. In (\ref{['fig5:densidadciudaddensa']}) and (\ref{['fig5:Unormaciudaddensa']}) we can observe how, if buildings cover more area of the city, drivers must advance slow, reaching only low velocities, and taking more time to approach the city center after gathering around obstacles. Unlike this, as shown in (\ref{['fig5:densidadciudaddispersa']}) and (\ref{['fig5:Unormaciudaddispersa']}), an area with more streets facilitates the traffic flow, allowing higher velocities, and spending less time to reach the city center.