Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part II: Degenerate Collision Frequency

Crouseilles, Nicolas; Hivert, Hélène; Lemou, Mohammed

doi:10.1137/15m1053190

Cited by 12 publications

(15 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In the near future, we aim at extending this work to more general context, considering higher dimensions, non periodic boundary conditions. The case of singular collision frequency also may generate anomalous diffusion (see [1]) and this also deserves a numerical study that we plan to do in a forthcoming work [8].…”

Section: Discussionmentioning

confidence: 98%

Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part I: The Case of Heavy-Tailed Equilibrium

Crouseilles¹,

Hivert²,

Lemou³

2016

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 98%

Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part I: The Case of Heavy-Tailed Equilibrium

Crouseilles¹,

Hivert²,

Lemou³

2016

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We proceed similarly for Case 2 (degenerate collision frequency), except that we perform the change of variable w = ε|k|v/ν(v) at the same places. It is important to note that the discretizations in velocity are independent of ε and we still have the first order (in time) uniform accuracy with respect to ε; this statement is proved in [4].…”

Section: Duhamel Formulation Based Schemementioning

confidence: 75%

“…This scheme is of order 1 uniformly in ε: ∃C > 0 independent from ε, such that max k |ρ ν (t n , k)−ρ n ν (k)| ≤ C∆t, ∀0 ≤ n ≤ N, N ∆t = T . We refer to [3] and [4] for more details and for the proof of the uniform accuracy of this scheme.…”

Section: Duhamel Formulation Based Schemementioning

confidence: 99%

Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit

Crouseilles

Hivert

Lemou

2015

Comptes Rendus. Mathématique

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…Thus, solving numerically the equivalent micro-macro formulation instead of the perturbed kinetic system will permit shifting automatically the limit problem, if the perturbation parameter (Knudsen parameter or sometimes referred to as the mean free path) is too small. Several contributions have investigated the asymptotic limit in the following cases: diffusion limit, [19][20][21][22] anomalous diffusion limit, 23,24 hyperbolic model, 25 and the Keller-Segel models of pattern formation in biological tissues. [26][27][28][29][30] Note that there are different approaches to construct such scheme for kinetic models in various contexts.…”

Section: Introductionmentioning

confidence: 99%

Kinetic‐fluid derivation and mathematical analysis of the cross‐diffusion–Brinkman system

Bendahmane

Karami

Zagour

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we propose a new nonlinear model describing the dynamical interaction of two species within a viscous flow. The proposed model is a cross-diffusion system coupled with the Brinkman problem written in terms of velocity fluid, vorticity, and pressure and describing the flow patterns driven by an external source depending on the distribution of species. In the first part, we derive macroscopic models from the kinetic-fluid equations by using the micro-macro decomposition method. On the basis of the Schauder fixed-point theory, we prove the existence of weak solutions for the derived model in the second part. The last part is devoted to developing a one-dimensional finite volume approximation for the kinetic-fluid model, which is uniformly stable along the transition from kinetic to macroscopic regimes. Our computation method is validated with various numerical tests. KEYWORDSBrinkman flows, cross-diffusion, finite volume method, kinetic theory, Schauder fixed-point theory 6288

show abstract

Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part II: Degenerate Collision Frequency

Cited by 12 publications

References 25 publications

Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part I: The Case of Heavy-Tailed Equilibrium

Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part I: The Case of Heavy-Tailed Equilibrium

Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit

Kinetic‐fluid derivation and mathematical analysis of the cross‐diffusion–Brinkman system

Contact Info

Product

Resources

About