A multi-layer model for self-propelled disks interacting through alignment and volume exclusion

We present an individual-based model describing disk-like self-propelled particles moving inside parallel planes. The disk directions of motion follow alignment rules inside each layer. Additionally, the disks are subject to interactions with those of the neighboring layers arising from volume exclusion constraints. These interactions affect the disk inclinations with respect to the plane of motion. We formally de-rive a macroscopic model composed of planar Self-Organized Hydrodynamic (SOH) models describing the transport of mass and evolution of mean direction of motion of the disks in each plane, supplemented with transport equations for the mean disk inclination. These planar models are coupled due to the interactions with the neighboring planes. Numerical comparisons between the individual-based and macroscopic models are carried out. These models could be applicable, for instance, to describe sperm-cell collective dynamics

Finite Volume approximations of the Euler system with variable congestion

We are interested in the numerical simulations of the Euler system with variable congestion encoded by a singular pressure. This model describes for instance the macroscopic motion of a crowd with individual congestion preferences. We propose an asymptotic preserving (AP) scheme based on a conservative formulation of the system in terms of density, momentum and density fraction. A second order accuracy version of the scheme is also presented. We validate the scheme on one-dimensional test-cases and extended here to higher order accuracy. We finally carry out two dimensional numerical simulations and show that the model exhibit typical crowd dynamics

Méthodes asymptotico-numériques pour des problèmes issus de la physique des plasmas et de la modélisation des interactions sociales

In this thesis, we devise analytical and numerical methods for capturing the asymptotic dynamics of plasma physics problems and collective movement models for animal populations. In the first part, we present a Particle-In-Cell numerical method for the Vlasov-Poisson system that is asymptotic preserving for the quasineutral limit. In the second part, we study the macroscopic limit of a Vicsek model that describes alignement interactions among two populations: a moving population and a steady one. Then we select a numerical scheme for capturing the solutions of the macroscopic Vicsek model corresponding to the underlying particle dynamics. The third part is dedicated to the incompressible-compressible transitions that appear in a macroscopic model for collective displacements with congestion effects. Asymptotic preserving numerical schemes for the congestion limit are then built for the Euler system with a maximal density constraint.Dans cette thèse, nous développons des méthodes analytiques et numériques pour capturer les dynamiques asymptotiques de problèmes issus de la physique des plasmas et de la modélisation des mouvements collectifs dans les populations animales. Dans une première partie, nous présentons une méthode numérique Particle-In-Cell (PIC) pour le système Vlasov-Poisson préservant l'asymptotique quasi-neutre. Dans une seconde partie, nous étudions la limite macroscopique d'un modèle de Vicsek décrivant des interactions d'alignement entre deux populations, une population à l'arrêt et une population en mouvement. Nous sélectionnons ensuite un schéma numérique pour capturer les solutions du modèle macroscopique de Vicsek correspondant à la dynamique particulaire sous-jacente. La troisième partie est dédiée à l'étude des transitions compressible-incompressible apparaissant sous l'effet d'une contrainte de congestion dans un modèle macroscopique de déplacement collectif. Des schémas numériques préservant l'asymptotique de congestion sont ensuite mis au point pour le système d'Euler avec une contrainte de densité maximale

High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation

We construct a high order discontinuous Galerkin method for solving general hyperbolic systems of conservation laws. The method is CFL-less, matrix-free, has the complexity of an explicit scheme and can be of arbitrary order in space and time. The construction is based on: (a) the representation of the system of conservation laws by a kinetic vectorial representation with a stiff relaxation term; (b) a matrix-free, CFL-less implicit discontinuous Galerkin transport solver; and (c) a stiffly accurate composition method for time integration. The method is validated on several one-dimensional test cases. It is then applied on two-dimensional and three-dimensional test cases: flow past a cylinder, magnetohydrodynamics and multifluid sedimentation

A two-species hydrodynamic model of particles interacting through self-alignment

In this paper, we present a two-species Vicsek model, that describes alignment interactions of self-propelled particles which can either move or not. The model consists in two populations with distinct Vicsek dynamics that interact only via the passage of the particles from one population to the other. The derivation of a macroscopic description of this model is performed using the methodology used for the Vicsek model: we find out a regime where alignment in the whole population occurs. We obtain a new macroscopic model for the densities of each populations and the common mean direction of the particles. The treatment of the non-conservativity of the interactions requires a detail study of the linearised interaction operator

A low cost semi-implicit low-Mach relaxation scheme for the full Euler equations

We introduce a semi-implicit two-speed relaxation scheme to solve the compressible Euler equations in the low Mach regime. The scheme involves a relaxation system with two speeds, already introduced by Bouchut, Chalons, Guisset (2019) in the barotropic case. It is entropy satisfying and has a numerical viscosity well-adapted to low Mach flows. This relaxation system is solved via a dynamical Mach number dependent splitting, similar to the one proposed by Iampietro et al. (2018). Stability conditions are derived, they limit the range of admissible relaxation and splitting parameters. We resolve separately the advection part of the splitting by an explicit method, and the acoustic part by an implicit method. The relaxation speeds are chosen so that the implicit system fully linearizes the acoustics and requires just to invert an elliptic operator with constant coefficients. The scheme is shown to well capture with low cost the incompressible slow scale dynamics with a timestep adapted to the velocity field scale, and rather well the fast acoustic waves

Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior -- an approximation of the steady solution -- which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions