Séminaire de mathématiques appliquées (archives)

Les modèles hydrodynamiques pour la fusion par confinement inertiel doivent être fermés en four- nissant une loi pour le flux de chaleur des électrons. Dans la plupart des cas hors-équilibre, la loi locale de Spitzer-Härm est insuffisante pour restituer l’ensemble des phénomènes physiques. En effet, la présence de forts gradients de température engendre l’apparition de flux de température non locaux qui rendent cette approche macroscopique incomplète. Pour restituer cet effet cinétique, la résolution d’une équation cinétique coûteuse à l’échelle microscopique serait requise. Néanmoins, du fait des situations physiques considérées, des modèles à l’échelle mésoscopique [1, 2] s’avèrent suffisants. En particulier, une approche aux moments permet de répondre à ces besoins de modélisation et de réduire le coût numérique ; d’autre part, l’utilisation d’un tel modèle hyperbolique, du fait de sa construction, présente l’avantage d’être suffisamment flexible pour y ajouter une physique plus complexe (champs magnétiques par exemple).

Dans ce travail, nous nous concentrons sur la résolution numérique du modèle M1 du transport ther- mique non local sans champ magnétique. La nature multi-échelle de ce modèle rend l’élaboration d’un schéma numérique difficile en termes de préservation de l’asymptotique pour capter les différents ré- gimes en fonction du nombre de Knudsen. Pour traiter ce problème, nous nous proposons d’utiliser UGKS (Unified Gas Kinetic Schema) [3, 4] ; un schéma robuste pour l’équation cinétique reposant sur la solution intégrale de l’équation cinétique pour élaborer les flux. Cette méthode présente l’avantage de préserver l’asymptotique de l’équation en résolvant correctement à la fois le régime non local associé à du transport (hyperbolique) et le régime local associé à de la diffusion (parabolique). Pour obtenir un schéma pour le modèle aux moments, une méthode générique est proposée dans laquelle le flux numérique d’UGKS est fermé avec la fonction de distribution M1. Cette technique revient à projeter la fonction de distribution dans l’espace M1 à chaque pas de temps dans UGKS.

Afin d’implémenter ce schéma, une méthode de quadrature pour calculer des demi-moments de fonction de distribution M1 sur la sphère est proposée. De plus, une extension à l’ordre 2 n’affectant pas la préservation de l’asymptotique est suggérée. La flexibilité de ce schéma est aussi démontrée dans sa capacité à dégénérer vers un schéma de diffusion arbitrairement choisi. Finalement, cette nouvelle méthode est validée et testée sur différents cas tests.

Hilbert, in his 1900 so-called "problems lecture", formulated as 6th problem the challenge to find an axiomatic basis for mechanics and probability. Kolmogorov's 1933 "Grundbegriffe" monograph was widely accepted as an adequate answer to this challenge regarding the axiomatisation of --- one has to say now --- "classical" probability. Coincidentally, 1900 is also the year when Planck formulated his thesis of energy quanta, which would give rise to quantum theory, and which requires a new probability theory. This became clear after Heisenberg's 1925 paper and his Göttingen colleagues' works afterwards, which together with Dirac introduced the algebraic point of view, culminating in von Neumann's 1932 monograph on the widely known Hilbert space representation of quantum mechanics; which actually appeared before Kolmogorov's monograph. The main difference between the classical and the "new" probability lay in the non-commutativity of random variables. Today there are also other areas where such quantum like behaviour (QLB) seems to occur.

The algebraic view offers a way on how to treat both classical and quantum like phenomena in a unified mathematical setting. And although probabilists today seem to be happy with Kolmogorov's approach based on measure theory, it may be interesting to look at the subject through a different pair of glasses. This algebraic view also offers a more direct way to address random variables with values in infinite dimensional spaces, something which with classical measure theory can only be done in a somewhat circumlocutory fashion. It also helps to separate purely algebraic questions from analytical ones, but of course thrives in the interplay of both.

Without wanting to present a strict axiomatic derivation, the start will be an early --- and in the light of modern theory also abstract algebraic --- view on random variables, as can be found implicitly in the work of early probabilists like the Bernoullis. Their properties are sketched as emanating from simple operational requirements regarding random variables, the mean or expectation, as well as sampling or observations. Concrete representations of this abstract setting connect it with algebras of linear mappings and the spectral theory of these, and one may recover Kolmogorov's classical characterisation as one particular representation.

Striking differences between classical or commutative probability and non-commutative probability appear already with simple linear algebra. As this is a subject which nowadays all engineering and science students learn at a very early stage, it may also be an interesting approach to teaching probability. And possible novel devices like quantum computers can be described in this setting.

This algebraic view has also a functional analytic extension, which can be used to construct generalised random variables and "ideal elements". It allows the specification of not only analogues of all the classical spaces of random variables, but to go beyond this and address questions of "smoothness" on the one hand, and the definition of idealised elements resp. "generalised" random variables on the other hand. This very much echoes the construction of distributions resp. generalised functions in the sense of Sobolev and Schwartz.

The method of moments is commonly used to reduce a kinetic equation into a fluid model. In this talk, I will present this technique as a semi-discretization with respect to the kinetic variable. I will focus on the main properties expected for this approximation, namely the positivity of an underlying kinetic approximation, a.k.a. the realizability, the strong or weak hyperbolicity and the entropy dissipation of the resulting system. I will present some novelties around these approximations, classified in three categories: the quadrature-based methods, the entropy-based methods and the realizability-based methods. Eventually, I will give some ideas on how to analyze such approximations and illustrate it on some kinetic toy problems.

In this paper, we study the asymptotic behavior of the global solution to a degenerate forest kinematic model, under the action of a perturbation modelling the impact of climate change. When the main nonlinearity of the model is assumed to be monotone, we prove that the global solution converges to a stationary solution, by showing that a Lyapunov function deduced from the system satisfies a Lojasiewicz-Simon gradient inequality. Under suitable assumptions on the parameters, we prove the continuity of the flow and of the stationary solutions with respect to the perturbation parameter. Although, due to a lack of compactness, the system does not admit the global attractor, we succeed in proving the robustness of the weak attractors, by establishing the existence of a family of positively invariant regions. We also present numerical simulations of the model and experiment the behavior of the solution under the effect of several types of perturbations. Finally, we show that the forest kinematic model can lead to the emergence of chaotic patterns