Macroscopic modeling and simulations of room evacuation

We analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne-Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost, which results in minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-finite element method to solve the problems and present error analysis for the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes' model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.Comment: 22 page

A numerical comparison between degenerate parabolic and quasilinear hyperbolic models of cell movements under chemotaxis

We consider two models which were both designed to describe the movement of eukaryotic cells responding to chemical signals. Besides a common standard parabolic equation for the diffusion of a chemoattractant, like chemokines or growth factors, the two models differ for the equations describing the movement of cells. The first model is based on a quasilinear hyperbolic system with damping, the other one on a degenerate parabolic equation. The two models have the same stationary solutions, which may contain some regions with vacuum. We first explain in details how to discretize the quasilinear hyperbolic system through an upwinding technique, which uses an adapted reconstruction, which is able to deal with the transitions to vacuum. Then we concentrate on the analysis of asymptotic preserving properties of the scheme towards a discretization of the parabolic equation, obtained in the large time and large damping limit, in order to present a numerical comparison between the asymptotic behavior of these two models. Finally we perform an accurate numerical comparison of the two models in the time asymptotic regime, which shows that the respective solutions have a quite different behavior for large times.Comment: One sentence modified at the end of Section 4, p. 1

A well-balanced numerical scheme for a one dimensional quasilinear hyperbolic model of chemotaxis

28 pagesInternational audienceWe introduce a numerical scheme to approximate a quasi-linear hyperbolic system which models the movement of cells under the influence of chemotaxis. Since we expect to find solutions which contain vacuum parts, we propose an upwinding scheme which handles properly the presence of vacuum and, besides, which gives a good approximation of the time asymptotic states of the system. For this scheme we prove some basic analytical properties and study its stability near some of the steady states of the system. Finally, we present some numerical simulations which show the dependence of the asymptotic behavior of the solutions upon the parameters of the system

Modélisation du transport de plasmides d'ADN du milieu extracellulaire au noyau par électroporation.

We propose a mathematical model for the DNA plasmids transport from the extracellular matrix up to the cell nucleus. The model couples two phenomena: the electroporation process, describing the cell membrane permeabilization to plasmids and the intracellular transport enhanced by the presence of microtubules. Numerical simulations of cells with arbitrary geometry and a network of microtubules show numerically the importance of the microtubules and the electroporation on the effectiveness of the DNA transfection, as observed by previous biological data.Le but de ce rapport est de présenter un modèle mathématique pour le transport de plasmides d'ADN, du milieu extracellulaire jusqu'au noyau, par application d'un champ électrique électroporant. Le modèle couple deux phénomènes : le processus électrique d'électroperméabilisation de la membrane cellulaire, ainsi que le transport électrophorétique de l'ADN dans le milieu extracellulaire d'une part, et le processus de transport actif de l'ADN le long des microtubules d'autre part. Les simulations numériques démontrent l'importance du transport actif le long des microtubules ainsi que l'avantage de l'électroporation pour la transfection de gènes, ce qui est corroboré par de récentes données expérimentales

Numerical approximation and analysis of mathematical models arising in cells movement

The Phd thesis is devoted to the numerical and mathematical analysis of systems of partial differential equations arising in the modeling of cells movement. The model of nutrient-dependent tumour growth is built and the asymptotic stability of constant steady states for small perturbations is proved. Then the parabolic and hyperbolic models of chemotaxis are approximated using finite differences and finite volume methods. In particular, a consistent scheme, which is well-balanced on steady states with constant velocity, preserves the non negativity of the density and treats the vacuum, is constructed. Having efficient and accurate numerical methods the behaviour of the solutions is analyzed. At first the pure diffusion problem, for which the waiting time phenomenon and regularity under the physical boundary condition are of main concern. Then the attention is focused on the study of existence and stability of non constant stationary solutions and long time behaviour of the hyperbolic model of chemotaxis on a bounded domain.La thèse est d'éditée a l'analyse numérique et mathématique de systèmes d'équations aux dérivées partielles provenant de la modélisation du mouvement des cellules. Nous constructions un modèle de croissance tumorale dépendant des nutriments et nous montrons la stabilité asymptotique des états stationnaires constants pour des perturbations petites. Puis, les modèles paraboliques et hyperboliques de chimiotactisme sont approchés en utilisant des méthodes de différences finis et de volumes finis. En particulier, nous constructions un schéma consistant, le schéma well-balanced pour les états stationnaires a vitesse constante, qui conserve la positivité de la densité et traite le vide. Avec ces méthodes numériques efficaces et précises, le comportement des solutions est analysé. Premièrement le problème de la diffusion pure est étudié et le phénomène du temps d'attente et la régularité sous la condition physique aux bords sont considérés. Puis nous concentrons notre attention sur l'étude de l'existence et de la stabilité des états stationnaires et sur le comportement en temps long du modèle du chimiotactisme sur un domaine borné

Travelling Chemotactic Aggregates at Mesoscopic Scale and BiStability

International audienceWe investigate numerically a model consisting in a kinetic equation for the biased motion of bacteria following a run-and-tumble process, coupled with two reaction-diffusion equations for chemical signals. This model exhibits asymptotic propagation at a constant speed. In particular, it admits travelling wave solutions. To capture this propagation, we propose a well-balanced numerical scheme based on Case's elementary solutions for the kinetic equation, and L-splines for the parabolic equations. We use this scheme to explore the Cauchy problem for various parameters. Some examples far from the diffusive regime lead to the coexistence of two waves travelling at different speeds. Numerical tests support the hypothesis that they are both locally asymptotically stable. Interestingly, the exploration of the bifurcation diagram raises counter-intuitive features

Comparative Study of Macroscopic Pedestrian Models

International audienceWe analyze numerically some macroscopic models of pedestrian motion to compare their capabilities of reproducing characteristic features of crowd behavior, such as travel times minimization and crowded zones avoidance, as well as complex dynamics like stop-and-go waves and clogging at bottlenecks. We compare Hughes’ model with different running costs, a variant with local dependency on the density gradient proposed in Xia et al. (2009), and a second order model derived from the Payne-Whitham traffic model which has first been analyzed in Jiang et al. (2010). In particular, our study shows that first order models are incapable of reproducing stop-and-go waves and blocking at exits