Special Session 187: New Trends in Crowd Dynamics and Traffic Flow

Congested Crossing Pedestrian Traffic Flow : Dispersion vs. Transport in Crowded Areas

Mariam AL KHATIB University of Limoges France Co-Author(s):

Abstract: This study investigates the dynamic interactions between two populations coexisting within a shared spatial domain. We develop theoretical and numerical approaches to analyze situations in which one population must traverse a territory occupied by another, requiring strategies to mitigate congestion due to spatial limitations. Population~1 is modeled through a linear transport equation, while Population~2 is described by a granular diffusion model inspired by sandpile dynamics, capturing its internal reorganization and natural tendency to decongest. Numerical simulations are used to explore how different movement strategies adopted by the traversing population---such as directed motion toward a target, internal spreading to reduce local density, or uniform dispersion across the domain---influence the response and redistribution of the resident population. The results highlight how the interplay between transport and granular diffusion mechanisms shapes the global behavior of the system.

Error Estimate for a Semi-Lagrangian Scheme for a Time-Dependent Hamilton-Jacobi Equation on Networks

Elisabetta Carlini Sapienza University of Rome Italy Co-Author(s):

Abstract: We study the numerical approximation of time-dependent Hamilton-Jacobi equations on networks. In particular, we introduce a semi-Lagrangian scheme and prove a convergence error estimate of order 1/2. The analysis relies on showing the equivalence between two notions of solutions proposed by Imbert and Monneau (2017) and Siconolfi (2022). Numerical simulations illustrating the performance of the scheme are also presented.

Deterministic Mean Field Games with jumps

Annette Dumas Universit\`e de Limoges, XLIM France Co-Author(s):    Filippo Santambrogio

Abstract: The Mean Field Game we consider is motivated by the modelization of the housing dynamics where each inhabitant can move from one place to another. In particular, the trajectories of the agents are piecewise constant and they minimize a cost consisting in the number of jumps (or relocations) and two terms depending on the density: the first one is variational and the other one is non-variational. A Nash equilibrium for this mean field game is a measure over the curves minimizing a problem in a Lagrangian form which depends on the measure itself. To prove the existence of a Nash equilibrium, we reformulate the problem, thanks to an optimal transport result, in a Eulerian form for which we prove regularity results. The Eulerian formulation also allows us to perform numerical simulations thanks to a Fast Dual Proximal Gradient method for which the convergence of the iterations is guaranteed by Beck and Teboulle (2014). We will see that the mixed structure with the non-variational term in the problem enables us to consider more complex cost functions and also examples with two populations.

On a modified Hughes model for crowd motion

Hamza Ennaji Univ. Grenoble Alpes, CNRS, Grenoble INP*, LJK France Co-Author(s):    Noureddine Igbida, Jos\`{e} Miguel Urbano

Abstract: We introduce a macroscopic crowd motion model inspired by Hughes (2002, 2003), coupling a nonlinear conservation law for pedestrian density with an Eikonal equation characterizing the shortest path. This approach modifies Hughes` original formulation and refines the prediction-correction framework recently proposed by Ennaji, Igbida and Jradi (2023). By incorporating anticipatory behavior and dynamic route adjustment, our model offers a realistic representation of crowd dynamics in complex environments. Finally, we present the mathematical formulation, discuss its well-posedness, and illustrate its qualitative behavior through numerical simulations.

A New Class of Non-Linear PDEs for Species Dispersal under Congestion Constraints

Noureddine igbida Unniveristy of Limoges France Co-Author(s):

Abstract: This talk introduces a new class of cross-diffusion systems designed to study the dispersal of species under overcrowding constraints. Moving beyond classical $W_2-$Wasserstein flows or standard PDE couplings, we propose an approach based on proximal energy minimization through a minimum flow process. This framework allows us to establish a well-posed PDE system that captures the delicate interplay between diffusion and concentration gradients. Specifically, for homogeneous cases, we derive a well-defined PDE grounded in a novel $H^{-1}-$theory specifically developed for overcrowding dispersal.

Fully discrete follow-the-leader approximation of one-dimensional scalar conservation laws with vacuum

Valeria Iorio University of L`Aquila Italy Co-Author(s):

Abstract: We present a fully discrete particle approximation for one-dimensional scalar conservation laws. Under suitable monotonicity assumptions on the macroscopic velocity, we construct a vacuum-compatible family of time-discrete particle equations and show that an appropriate piecewise-constant density reconstruction from the particle setting converges to the unique entropy weak solution of the macroscopic scalar conservation law. This is a joint work with Marco Di Francesco, Simone Fagioli, and Massimiliano Daniele Rosini.

Analysis of some pedestrians PDE models for a population under stress

Kamal Khalil LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France. France Co-Author(s):

Abstract: In this talk, we investigate several macroscopic PDE models for pedestrian dynamics describing the spatio-temporal evolution of a population under stress (panic) in dangerous situations. We first present a first-order model for the evacuation of a stressed population from a room with an exit, incorporating different interacting human behaviors. For this model, we establish the local existence, uniqueness, and regularity of solutions using semigroup theory, together with their positivity and \(L^1\)-boundedness. To illustrate the propagation of stress (panic) and its impact on evacuation dynamics, we provide numerical simulations of several evacuation scenarios, with particular emphasis on populations with a low-risk culture in emergency situations. We then introduce a second-order model to better capture stress effects within the population and compare its dynamics with those of the first-order model.

Merging agent-based and Mean Field Game descriptions of crowd dynamics

Konstantinos Koutsomitis Universite Paris Saclay/ LPTMS France Co-Author(s):    C\`{e}cile Appert-Rolland, Denis Ullmo

Abstract: Mean Field Games is a powerful tool to describe competitive optimization processes among agents. A natural field of application is crowd dynamics where pedestrians optimize their path, while interacting with the others. Due to the backward-forward structure of the MFG equations, anticipation is included in the model, which is often a key element to reproduce pedestrian behavior. In this formalism, the interactions among the agents are usually considered exclusively inside the minimization of a functional (personal cost), in the decision-making level and not in the operational, at the agent scale. This approach often fails to describe situations where the granular effects are dominant, namely while passing through a bottleneck. To enhance the realism of the framework, we aim to incorporate these granular effects by considering explicit microscopic interactions in the Mean Field Game derivation. In this direction, we treat a toy-model by replacing Langevin dynamics for the agents with a one-dimensional Exclusion Process, which is a paradigmatic model of interacting particles, especially in transport and congestion phenomena. We will show how to derive the corresponding Mean Field Game equations and discuss the effect of these interactions on the Nash equilibrium of the game, in particular in the case of evacuations. This study can be seen as a first step towards the study of other interacting models and their potential Mean Field Games counterparts.

Fully discrete methods for Wasserstein gradient flows and crowd motion

Filippo Santambrogio Institut Camille Jordan France Co-Author(s):    Anatole Gallouet, Anastasiia Hraivoronska

Abstract: In 2010 we introduced, together with Maury and Roudneff-Chupin, a simple evolution model for crowd motion which happens to be a gradient flow in the Wasserstein space $W_2$ of a potential energy with a density constraint. Recently, with Hraivoronska we proposed a full discretization of some Wasserstein gradient flows on a fixed grid and found the necessary and sufficient condition in terms of the time and space steps to obtain convergence. This method is particularly efficient in the case of the crowd motion gradient flow, as it only requires to solve a sequence of linear programming problems. In a joint ongoing work with A. Gallou\et and Hraivoronska we exploit this in order to obtain efficient numerical simulations: I will present these results and all the strategies that we develop to reduce the corresponding computational cost.

Optimal Control of Sweeping Processes: Theoretical Framework and Numerical Approximation

Francisco Silva XLIM, Universite de Limoges France Co-Author(s):    Justina Gianatti and Emilio Vilches

Abstract: In this talk, based on joint work with J. Gianatti and E. Vilches, we first provide a theoretical framework for finite horizon optimal control problems in which the state is governed by a controlled sweeping process, and we show that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. We then introduce a semi-Lagrangian scheme to approximate the value function and establish its convergence. Finally, we present numerical experiments for two-dimensional problems and discuss some promising extensions.

From Mean Field Games to a Hughes-Type Model: A Lagrange-Galerkin Approach

Ahmad Zorkot University of Vienna Austria Co-Author(s):    Elisabetta Carlini and Francisco J. Silva

Abstract: We present a numerical scheme for a Hughes-type model where agents optimize based only on the current population distribution $m(t)$. Unlike classical Mean Field Games where agents anticipate future crowd evolution, at each time step a new Hamilton-Jacobi-Bellman equation is solved with the instantaneous density. Our method combines a semi-Lagrangian discretization for the value function with a Lagrange-Galerkin approximation for the continuity equation. We prove well-posedness and convergence of the scheme. We also introduce a hybrid system coupling both MFG and Hughes-type populations. Various numerical tests are presented for both models, showing distinct behavioral patterns in congestion and target accumulation.