Numerical methods for asymptotic problems

Asymptotic-Preserving (AP) methods

The concept of an Asymptotic-Preserving (AP) method makes a breakthrough in the numerical resolution of asymptotic perturbations of Partial-Differential Equations. It has first been introduced by Shi Jin for transport equations in diffusive regimes (see S. Jin, SIAM J. Sci. Comp. 21, 441-454, 1999). ...

 

Quasineutral limit in plasmas

Strongly anisotropic diffusion problems ; application to large

magnetic fields in plas

mas 

 

Multi-scale methods

Multi-scale methods are needed when detailed information about the microscopic structure of the solution (i.e. the structure of the solution at scale 'eps') are needed. This may occur for instance when nonlinear interactions give rise to macroscopic structures which persist in the limit eps -> 0 .

Localized Model-Upscaling (LMU) and Macro-guided micro (MGM) methods

The localized model-upscaling (LMU) method consists in coupling a perturbation model and its asymptotic limit model (when the perturbation parameter is sent to zero) through a transition zone. ...

Multi-Scale Finite-Element Method (MS-FEM)

This projects investigates the applicability of the Multi-Scale Finite-Element Method (MS-FEM) of T. Hou to diffusion equations in perforated domains. One targetted application is pollutant dispersion in cities. Pollutant dispersion is extremely dependent of the geometry of the city but its full account leads to very time consuming simulations. The MS-FEM method is able to provide real-time responses to critical events, which is extremely useful in crisis management. ...

 

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Emergence and self-organization in complex systems

Complex systems are characterized by the spontaneous formation of spatio-temporal structures as a result of simple local interactions between agents without leaders. Complex systems mostly appear in the biological and social contexts but can also be encountered in physics, chemistry, etc. The project aims at understanding the emergence of self-organization  through  the investigation of global, macroscopic models. ...

 

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Micro-macro passage in models of complex systems

Complex systems are commonly investigated through microscopic, Agent (or Individual)-Based Models (ABM or IBM), which predict the evolution of each agent in time. When macroscopic models are considered in the literature, they are usually based on phenomenological considerations.This project aims at rigorously establishing macroscopic models from their microscopic (Agent-Based Model) counterpart. By doing so, macroscopic models gain in predictive character and may become an invaluable aid for experimental data analysis. ...

 

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Self-Organized Hydrodynamics

The  'Self-Organized-Hydrodynamics' (SOH) model describes self-organization phenomena at the macroscopic level. It shows some prominent differences from usual compressible hydrodynamic models. For instance, it is non-conservative and involves a constraint on the flow speed. The SOH model is one of the very first examples of non-conservative hyperbolic systems that can be derived from kinetic theory. Its derivation highlights one of the specific difficulties in complex systems, namely that elementary particle interactions may lack conservation properties. ...

 

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Phase transitions in self-organized dynamics

In kinetic models of self-organized dynamics inspired from the Vicsek model, the distribution of particle orientations exhibits multiple 'Local Thermodynamical' equilibria (LTE): ...

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Propagation of chaos

The derivation of macroscopic models relies on the 'propagation of chaos', i.e. the fact that the particles become statistically independent when their number becomes large. In complex systems, self-organization may prevent propagation of chaos to arise. ...

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Emergence

Emergence refers to the ability of complex systems to spontaneously form large-scale structures which are not directly encoded in the agents' interaction rules. In models, emergence results from various mechanisms. In this page are found examples of

• positive feedback (in network formation models such as ant-trail networks)
• phase transitions
• unilateral constraints (such as the non-overlapping constraint)

  
  

Formation of structures through non-overlapping constraints

The typical phenomenology is that of a traffic jam: in the congested phase, the vehicle density is maximal, corresponding to the minimal allowed distance between two vehicles, while in the free flow phase, the vehicle density is below this threshold. Pedestrian crowds or mammalian herds are two-dimensional analogs of traffic jams. The density of individuals cannot exceed a maximal density threshold corresponding to the individuals being in contact with each other. When the density is below this threshold, the flow is compressible. When the density reaches the threshold, the flow becomes incompressible. ...

 

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Applications  

Fish schools

Experimental measurements of fish trajectories (J. Gautrais et al, J. Math. Biol. (2009) 58: 429-445) suggest that individual fish behaviour follows the 'Persistent Turning Walker' (PTW) process. The PTW model assumes that individual trajectories consist of pieces of circles with randomly varying radii, which are patched together (see picture below).  ...

 

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Models of network formation

Self-organized dynamics often leads to the organization of agents into networks. An Individual-Based Model of network generation by self-propelled particles has been developed. Each agent deposits directed pheromones and interacts with them through alignment interaction.   ...

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Herds

Herds of gregarious mammals offers a natural laboratory for the observation of emergence phenomena. Like in vehicular traffic or crowds, the spatial extension of the agents gives rise to congestion phenomena. Congestion phenomena and their morphogenetic capability within herds have been studied and simulated ...

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Pedestrians and crowds

Interactions between pedestrians in crowds produce emerging self-organized structures. In this work, we have studied this phenomenon under controlled laboratory conditions, in the framework of the 'PEDIGREE' ANR project. We have developed both microscopic (Individual-Based) and macroscopic (fluid-like) models. ...

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Collective cell dynamics

Collective cell dynamics is of key importance in such processes as embryogenesis, tissue regeneration or tumor growth. Tumors grown in vitro in 3D (or tumor spheroids, see IP3D web page and pictures) offer a fascinating laboratory system. A microscopic, individual model of cell growth, division and motion is under development...

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Other topics 

Other works concerning the following topics can be found on the 'publications' page  

• Micro-macro passage in physical systems
• Plasmas (general)
• Energy-transport models for plasmas and semiconductors
• Fokker-Planck-SHE (Spherical Harmonics Expansion) models for plasmas and semiconductors
• Diffusion and Hydrodynamic limits
• Diffusion by solid boundaries: applications to plasmas
• Diffusion by interfaces: applications to semiconductor super-lattices
• High-field diffusion models
• Quantum systems
• Relativistic systems
• Wave-particle collisions and applications to cometary flows and turbulence
• Polymers
• Homogenization
• Homogenization in rarefied gases and Knudsen pumps
• Homogenization in quantum systems and Einstein rate equations
• Schrödinger-Poisson systems with scattering states or Wigner-Poisson systems
• Entropic numerical methods for Boltzmann and Fokker-Planck Landau operators
• Child-Langmuir law
• Hyperbolic systems (hydrodynamics, Maxwell, etc.): theory and numerics
• Particle methods
• Mathematical theory of Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck

 

 

 

 

 

 

 

Table des matières

1. 1 Numerical methods for asymptotic problems
   1. 1.1 Asymptotic-Preserving (AP) methods
      1. 1.1.1 Quasineutral limit in plasmas
      2. 1.1.2 Strongly anisotropic diffusion problems ; application to large magnetic fields in plas mas 
      3. 1.1.3 All-speed scheme for compressible fluids, multiphase flows and fluids with density constraints
   2. 1.2 Multi-scale methods
      1. 1.2.1 Localized Model-Upscaling (LMU) and Macro-guided micro (MGM) methods
      2. 1.2.2 Multi-Scale Finite-Element Method (MS-FEM)
2. 2 Emergence and self-organization in complex systems
   1. 2.1 Micro-macro passage in models of complex systems
      1. 2.1.1 Self-Organized Hydrodynamics
      2. 2.1.2 Phase transitions in self-organized dynamics
      3. 2.1.3 Propagation of chaos
   2. 2.2 Emergence
      1. 2.2.1 Formation of structures through non-overlapping constraints
   3. 2.3 Applications  
      1. 2.3.1 Fish schools
      2. 2.3.2 Models of network formation
      3. 2.3.3 Herds
      4. 2.3.4 Pedestrians and crowds
      5. 2.3.5 Collective cell dynamics
3. 3 Other topics