Journées savoisiennes de mathématiques appliquées

METHODES PARTICULAIRES
DE SIMULATION NUMERIQUE
 

Le Bourget-du-Lac, 17 & 18 Mai 2001

Les méthodes particulaires sont utilisées pour résoudre des problèmes en dimension élevée ou posés dans des domaines à géométrie compliquée. Elles peuvent être stochastiques, comme les méthodes de Monte Carlo, ou déterministes. A cette dualité de méthodes correspond une double modélisation des phénomènes: par des équations aux dérivées partielles ou par des processus aléatoires. Il y a en général peu de communication entre ces deux types de modélisation et les deux familles de méthodes numériques associées. Le but de ce colloque est de réunir des spécialistes des deux descriptions et des deux approches algorithmiques d'une même réalité.

Comité d'organisation: Pierre BARAS, Georges-Henri COTTET, Christian LECOT
 


Liste des conférenciers
 
 

Fabrice DELUZET (Univ. Toulouse III)
Andreas EIBECK (W.I.A.S. Berlin)
Arturo KOHATSU (U.P.F. Barcelona)

Michael MASCAGNI (Florida State Univ.)
Sylvie MAS-GALLIC (Univ. d'Evry)
Shigeyoshi OGAWA (Univ. de Kanazawa)
Benoît PERTHAME (E.N.S. Paris)
Marco PICASSO (E.P.F. Lausanne)
Philippe PONCET (Univ. Grenoble I)
Denis TALAY (I.N.R.I.A. Sophia-Antipolis)


Programme

Jeudi 17 Mai

09 H 00 - 10 H 00  D. TALAY: Convergence rate analysis of stochastic particle methods for McKean Vlasov equations: a survey
10 H 00 - 11 H 00  A. KOHATSU: Malliavin Calculus and Weak Approximations
11 H 00 - 12 H 00  S. MAS-GALLIC: Une méthode particulaire déterministe à poids fixes pour des problèmes diffusifs

14 H 00 - 15 H 00  F. DELUZET: Implicit Particle in Cell method in fluid plasma simulations
15 H 00 - 16 H 00  S. OGAWA: Brownian particle equation and its applications
16 H 00 - 17 H 00  M. MASCAGNI: New Monte Carlo Methods for Problems in Materials and Biology
17 H 00 - 17 H 30  P. PONCET: Méthodes Particulaires Hybrides et Equations de Navier-Stokes en Domaine Cylindrique

Vendredi 18 Mai

09 H 00 - 10 H 00  B. PERTHAME: ES-BGK and Boltzmann models for dilute flows: theory and numerical comparisons
10 H 00 - 11 H 00  M. PICASSO: A finite element/Monte Carlo method for dilute polymer solutions
11 H 00 - 12 H 00  A. EIBECK: Stochastic particle approximation of the coagulation equation and gelation phenomena

Les conférences seront données en salle 180 de l' I.U.T.
voir le plan du site : http://www.lama.univ-savoie.fr/sitelama/fichiers_html4/route.html


Inscription

Les chercheurs intéressés sont invités à s'inscrire avant le 1er Mai 2001 à l'adresse

Secrétariat JSMA, Laboratoire de Mathématiques
Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France
Télécopie : 33 (0)4 79 75 81 42 - Téléphone : 33 (0)4 79 75 87 20
 


Résumés des conférences


Dr. Fabrice DELUZET

Mathématiques pour l'Industrie et la Physique
Bât I R2 - Université Paul Sabatier - Toulouse III
118, route de Narbonne
F - 31062 TOULOUSE CEDEX 4
FRANCE
téléphone: 33 (0)5 61 55 83 14
télécopie: 33 (0)5 61 55 83 65
mél: deluzet@mip.ups-tlse.fr
http://mip.ups-tlse.fr

Implicit Particle in Cell method in fluid plasma simulations

This communication is dedicated to plasma simulation. The plasma is represented by a two-fluid model : hydrodynamics equations are considered for ions, while electrons are associated an energy-transport model, in which inertia is dropped. This approximation is motivated by the large ratio of ionic to electronic masses which suggests that electronic inertia can be neglected. The set of equations is supplemented with Maxwell's equations which drive the electromagnetic field evolution.
The space discretization is based on a Particle In Cell method which consists in the finite differentiation of the Maxwell's equations on a Cartesian mesh, and a discretization of fluids equations using particles. In order to relax stability constraints, an implicit coupling for electrons and Maxwell's equations is developed, while the coupling between conservation laws associated to the ionic fluid and Maxwell's equations remains explicit.
The ability of the scheme obtained to deal with implicit situations is illustrated on Plasma Opening Switches operating simulations. These devices are used to conduct high power currents and are characterized by the capability to switch (from the full conduction phase to a state where the switch does not conduct any current) in few dozens of nanoseconds. The mechanisms that are responsible for the opening relies on space charge formation and requires for this purpose a bi-fluid model. As density considered rises, explicit schemes simulation parameters go to small values in order to fulfill stability constraints, while our implicit schemes remains stable for large time steps and mesh size.


Dr. Andreas EIBECK

Weierstrass Institut für Angewandte Analysis und Stochastik
Mohrenstrasse 39
D - 10117 BERLIN
ALLEMAGNE
téléphone: 49 (0)30 20372 482
télécopie: 49 (0)30 2044 975
mél: eibeck@wias-berlin.de
http://www.wias-berlin.de/~eibeck

Stochastic particle approximation of the coagulation equation and gelation phenomena

The coagulation equation describes the time evolution of the average concentration of clusters sticking together at a rate given by a binary mass-dependent kernel. Sufficiently fast growing coagulation kernels lead to solutions exhibiting a loss of the initial total mass. This phenomenon can be interpreted as the formation of infinitely large clusters, the so-called gel.

The Marcus-Lushnikov process provides a stochastic model for the coagulation equation and is extended to fragmentation, efflux and source terms. We propose a new stochastic particle system derived from a transformed coagulation equation and present convergence results. Numerical investigations yield that the new particle system reduces the systematic and statistical error. In particular, it provides a tool for detecting the mass of the gel in the correct way.


Prof. Arturo KOHATSU

Departament d'Economia i Empresa
Universitat Pompeu Fabra
Ramon Trias Fargas, 25-27
SP - 08005 BARCELONA
ESPAGNE
téléphone: 34 935 42 2750
télécopie: 34 935 42 1746
mél: arturo.kohatsu@econ.upf.es
http://www.econ.upf.es/~kohatsu/

Malliavin Calculus and Weak Approximations

In the past few years I have developped an alternative approach to the already classical proofs of weak approximations using the integration by parts formula of Malliavin Calculus. I believe that this is a useful toolin the cases when one wants to prove some weak convergence rates for stochastic equations where there is some irregularity in one of the components while all the others are highly regular. We will survey these results and some of its applications to the determination of the optimal rate of convergence of a particle method to the solution of the Mc Kean-Vlasov equation.


Prof. Michael MASCAGNI

Department of Computer Science
Florida State University
203 Love Building
TALLAHASSEE, FL 32306-4530
ETATS-UNIS D'AMERIQUE
téléphone: 1 850 644 32 90
télécopie: 1 209 796 87 46
mél: mascagni@cs.fsu.edu
http://www.cs.fsu.edu/~mascagni

New Monte Carlo Methods for Problems in Materials and Biology

Probabilistic potential theory enables us to solve a large class of parabolic and elliptic partial differential equations using diffusion
techniques.  Here, we present new first- and last-passage Monte Carlo algorithms and show their utility in problems coming from materials science and biology.  These techniques exploit the fact that the first-passage probability function is the Green's function for the Dirchlet problem of the Laplace equation.  First-passage algorithms allow the rapid simulation of diffusion using analytic or simulation-based Green's functions in rather less complicated basic geometries. This permits consideration of more complicated real geometries made up as combinations of the simple geometries where Green's functions are available.  This new method is the extension of the well known "Walk on Spheres" method.   Harnessing these first-passage algorithms, we have developed the fastest algorithms known to compute:
- The fluid permeability in overlapping, nonoverlapping,  and polydispersed spherical models of random porous media
- The Solc-Stockmayer model with zero potential, a model of ligand binding
- The mean trapping rate of a diffusing particle in a domain of nonoverlapping spherical traps
- The effective conductivity for perfectly insulating, nonoverlapping spherical inclusions in a matrix of conductivity

In certain problems, such as that of computing the electrostatic charge distribution on a conductor, using the last-passage distribution is useful. Using these analogous last-passage algorithms, we have solved the test problem of computing the charge distribution on a circular two-dimensional disk in three dimensions.

Our plans for the future involve adding more surface Green's functions to our present set of known Green's functions, and the application of these techniques to more realistic problems in materials and biology.

This is joint work with Dr. Chi-Ok Hwang of Florida State University, and Dr. James Given of Angle, Inc.


Prof. Sylvie MAS-GALLIC

Département de Mathématiques
Université d'Evry Val d'Essonne
Boulevard F. Mitterrand
F - 91025 EVRY CEDEX
FRANCE
téléphone: 33 (0)1 69 47 74 24
télécopie: 33 (0)1 69 47 74 19
mél: masgalli@cmapx.polytechnique.fr
http://www.univ-evry.fr/labos/lami/maths

Une méthode particulaire déterministe à poids fixes pour des problèmes diffusifs : la méthode de vitesse de diffusion

Cette méthode qui a vu le jour en physique des plasmas a pourtant été bien davantage utilisée pour des calculs de mécanique des
fluides que pour des problèmes de plasma. Elle sera présentée sur quelques exemples d'équations diffusives, linéaires ou non mais conservatives (milieux poreux, chaleur, Navier-Stokes ainsi que Landau, un modèle collisionnel de plasma).

Quelques possibilités de cette méthode seront illustrées par des résultats numériques.


Prof. Shigeyoshi OGAWA

Laboratory of Stochastic Systems
Department of Information and Systems Engineering
Faculty of Engineering - Kanazawa University
2-40-20 Kodatsuno
KANAZAWA 920-8667
JAPON
téléphone: 81 76 234 4911
télécopie: 81 76 234 4908
mél: s_ogawa@t.kanazawa-u.ac.jp
http://www.kanazawa-u.ac.jp

Brownian particle equation and its applications

We are to give a short but self contained review of the theory of Brownian particle equation (BPE), a class of stochastic partial
differential equations (SPDE) containing the Gaussian white noise as coefficients, and the noncausal stochastic calculus on which the theory of the SPDE should be constructed. We will also give some applications of the BPE theory to problems in mathematical sciences.


Prof. Benoît PERTHAME

Département de mathématiques et applications
Ecole Normale Supérieure
45, rue d'Ulm
75230 Paris cedex 05, France
FRANCE
téléphone: 33 (0)1 44 32 20 36
télécopie: 33 (0)1 44 32 20 80
mél: Benoit.Perthame@ens.fr
http://www.dma.ens.fr/users/perthame/

ES-BGK and Boltzmann models for dilute flows: theory and numerical comparison

A number of methods has been tested to compute faster rarefied flows modelled by a Boltzmann equation. This is a difficult tasking being given that the problem is posed in the phase space. The purpose of this talk is to survey some of these attempts at INRIA/M3N.

As a first method we will describe a coupling strategy Boltzmann/Euler (using resp. random particles methods and finite volumes) that has been devised successfully and which is based on kinetic solvers.

Another direction is to use simpler models than the quadratic Boltzmann kernel, such as relaxation equations. The Ellipsoidal Statistical BGK model (also called Gaussian BGK model) is an improvment of the classical relaxation towards a local Maxwellian (BGK model). It has been devised in order to provide the right transport coefficients (viscosity and heat transfert) in the Navier-Stokes asymptotics. It was proposed in the 70's by Holway and independently by Cercignani. The question of the entropy property (H-Theorem) was left open because the model involves non convex combinations of the second order velocity moments and of the usual isotropic matrix built on the temperature, and this discarded the model.

In fact, and unexpectedly,  the entropy property for the ES-BGK model holds true. In this talk we will give a proof (relying on Brunn-Minkowski inequality) and we will recall the motivation and the construction of the model.

We will also give sharp numerical comparisons between the BGK, the ES-BGK and the Boltzmann models on realistic two-dimensional computations. They show that a phenomenon occurs, like in turbulence modelling, that on a given grid the approximate model turns out to give better results.

The case of gas mixtures is also particularly interesting with this respect.


Dr. Marco PICASSO

Chaire d'Analyse et de Simulation Numériques
Département de Mathématiques
Ecole Polytechnique Fédérale de Lausanne
CH - 1015 LAUSANNE
CONFEDERATION HELVETIQUE
téléphone: 41 21 693 42 47
télécopie: 41 21 693 43 03
mél: Marco.Picasso@epfl.ch
http://dmawww.epfl.ch/~picasso

A finite element / Monte Carlo method for dilute polymer solutions

A Finite Element/Monte Carlo procedure is considered for solving the flow of a diluted solution of polymers. The polymeric liquid is assumed to be a newtonian solvent plus non interacting polymer chains. The mass and momentum equations yield the incompressible Navier-Stokes equations, with a source term to account for the extra-stress due to the polymer chains. The polymer chains are modeled using dumbbells, that is two beads connected with an elastic spring. Stochastic differential equations are obtained for the springs elongations. The extra-stress is then computed from the expectance of theses elongations.

A numerical procedure is proposed. The velocity, pressure, extra-stress and springs elongations belong to standard Finite Elements spaces. A Monte Carlo method is used to approach the expectance. Variance reduction and numerical results are presented.


M. Philippe PONCET

Laboratoire de Modélisation et Calcul
Université Grenoble IB.P. 53
38041 GRENOBLE Cedex 9
téléphone: 33 (0)4 76 63 57 13
télécopie: 33 (0)4 76 63 12 63
mél: Philippe.Poncet@imag.fr
http://www-lmc.imag.fr/lmc-edp/Philippe.Poncet

Méthodes Particulaires Hybrides et Equations de Navier-Stokes en Domaine Cylindrique

Les méthodes particulaires hybrides utilisent un couplage grille-particule. Un tel couplage est intéressant pour les calculs
tridimensionnels, car il propose une alternative entre les méthodes probabilistes qui convergent lentement et les méthodes purement
lagrangiennes dont le coût de calcul est prohibitif en trois dimensions.

Cet exposé présente un algorithme à pas fractionnaire pour les équations de Navier-Stokes, avec une simulation de collision
anneau/cylindre comme exemple validatif. Puis le problème des sillages de cylindre avec ses instabilités tridimensionnelles sera
présenté, ainsi qu'un exemple de contrôle de force trainée.
 


Dr. Denis TALAY

Institut National de Recherche en Informatique et en Automatique
2004, route des Lucioles
B.P. 93
F - 06902 SOPHIA-ANTIPOLIS
FRANCE
téléphone: 33 (0)4 92 38 78 98
télécopie: 33 (0)4 92 38 75 50
mél: Denis.Talay@sophia.inria.fr
http://www-sop.inria.fr/omega/personnel/Denis.Talay/

Convergence rate analysis of stochastic particle methods for McKean Vlasov equations: a survey


Les organisateurs: Pierre.Baras@univ-savoie.fr  -  Georges-Henri.Cottet@imag.fr  -  Christian.Lecot@univ-savoie.fr