In this work, we propose some numerical schemes for linear kinetic equations in the diffusion and anomalous diffusion limit. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion type equation. However, when a heavy-tailed distribution is considered, another time scale is required and the small mean free path limit leads to a fractional anomalous diffusion equation. Our aim is to develop numerical schemes for the original kinetic model which works for the different regimes, without being restricted by stability conditions of standard explicit time integrators. First, we propose some numerical schemes for the diffusion asymptotics; then, their extension to the anomalous diffusion limit is studied. In this case, it is crucial to capture the effect of the large velocities of the heavy-tailed equilibrium, so that some important transformations of the schemes derived for the diffusion asymptotics are needed. As a result, we obtain numerical schemes which enjoy the Asymptotic Preserving property in the anomalous diffusion limit, that is: they do not suffer from the restriction on the time step and they degenerate towards the fractional diffusion limit when the mean free path goes to zero. We also numerically investigate the uniform accuracy and construct a class of numerical schemes satisfying this property. Finally, the efficiency of the different numerical schemes is shown through numerical experiments.
Horizontal gene Transfer (HT) denotes the transmission of genetic material between two living organisms, while the vertical transmission refers to a DNA transfer from parents to their offspring. Consistent experimental evidence report that this phenomenon plays an essential role in the evolution of certain bacterias. In particular, HT is believed to be the main instrument of developing the antibiotic resistance. In this work, we consider several models which describe this phenomenon: a stochastic jump process (individual-based) and the deterministic nonlinear integrodifferential equation obtained as a limit for large populations. We also consider a Hamilton-Jacobi equation, obtained as a limit of the deterministic model under the assumption of small mutations. The goal of this paper is to compare these models with the help of numerical simulations. More specifically, our goal is to understand to which extent the Hamilton-Jacobi model reproduces the qualitative behavior of the stochastic model and the phenomenon of evolutionary rescue in particular.
Abstract. In this work, which is the continuation of [9], we propose numerical schemes for linear kinetic equation which are able to deal with the fractional diffusion limit. When the collision frequency degenerates for small velocities it is known that for an appropriate time scale, the small mean free path limit leads to an anomalous diffusion equation. From a numerical point of view, this degeneracy gives rise to an additional stiffness that must be treated in a suitable way to avoid a prohibitive computational cost. Our aim is therefore to construct a class of numerical schemes which are able to undertake these stiffness. This means that the numerical schemes are able to capture the effect of small velocities in the small mean free path limit with a fixed set of numerical parameters. Various numerical tests are performed to illustrate the efficiency of our methods in this context.
We construct numerical schemes to solve kinetic equations with anomalous diffusion scaling. When the equilibrium is heavy-tailed or when the collision frequency degenerates for small velocities, an appropriate scaling should be made and the limit model is the so-called anomalous or fractional diffusion model. Our first scheme is based on a suitable micro-macro decomposition of the distribution function whereas our second scheme relies on a Duhamel formulation of the kinetic equation. Both are Asymptotic Preserving (AP): they are consistent with the kinetic equation for all fixed value of the scaling parameter ε > 0 and degenerate into a consistent scheme solving the asymptotic model when ε tends to 0. The second scheme enjoys the stronger property of being uniformly accurate (UA) with respect to ε. The usual AP schemes known for the classical diffusion limit cannot be directly applied to the context of anomalous diffusion scaling, since they are not able to capture the important effects of large and small velocities. We present numerical tests to highlight the efficiency of our schemes. To cite this article: N. Crouseilles, H. Hivert, M. Lemou, C. R. Acad. Sci. Paris, Ser. I 340 (2005) ???????. RésuméSchémas numériques multi-échelles pour leséquations cinétiques dans la limite de diffusion anormale.Nous construisons des schémas numériques pour résoudre leséquations cinétiques dans le régime de diffusion anormale. Lorsque l'équilibre présente une queue lourde ou lorsque la fréquence de collision dégénère pour les petites vitesses, un scaling approprié permet d'obtenir un modèle asymptotique appelé modèle de diffusion anormale ou fractionnaire. Le premier schéma que nous construisons est basé sur une décomposition micro-macro de la fonction de distribution tandis que le second s'appuie sur une formulation de Duhamel de l'équation de départ. Ces deux schémas sont Asymptotic Preserving (AP) : ils sont consistants avec l'équation cinétique lorsque le paramètre d'échelle ε > 0 est fixé et dégénèrent en un schéma consistant avec le modèle limite quand ε tend vers 0. Le deuxième schéma est même uniformément précis (UA) par rapportà ε. Les schémas AP qui sont connus dans le cas de la limite de diffusion classique ne peuvent pas directement s'appliquer au cas de la diffusion anormale car ils ne permettent de capturer les effets importants des petites et des grandes vitesses. Nous présentons des tests numériques pour mettre enévidence l'efficacité des schémas que nous présentons. Version française abrégéeLe but de ce travail est de mettre en place des schémas numériques pour leséquations cinétiques linéaires dans le cas de la limite de diffusion anomale. Quand la distribution d'équilibre est une fonctioǹ a queue lourde ou lorsque la fréquence de collision est dégénérée pour les petites vitesses, l'équation cinétique (1) tend vers l'équation dite de diffusion anormale (2) lorsque le paramètre d'échelle ε tend vers zéro. Dans cette limite, une raideur apparaît dans l'équation de départ et une résolution numérique dir...
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