Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium

Nicolas Crouseilles 1, 2 Hélène Hivert 1, 2 Mohammed Lemou 1, 2
1 IPSO - Invariant Preserving SOlvers
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
Abstract : 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. 1. Introduction. The modeling and numerical simulation of particles systems is a very active field of research. Indeed, they provide the basis for applications in neutron transport, thermal radiation, medical imaging or rarefied gas dynamics. According to the physical context, particles systems can be described at different scales. When the mean free path of the particles (i.e. the crossed distance between two collisions) is large compared to typical macroscopic length, the system is described at a microscopic level by kinetic equations; kinetic equations consider the time evolution of a distribution function which gives the probability of a particle to be at a given state in the six dimensional phase space at a given time. Conversely, when the mean free path is small, a macroscopic description (such as diffusion or fluid equations) can be used. It makes evolve macroscopic quantities which depends only on time and on the three dimensional spatial variable. In some situations, this description can be sufficient and leads to faster numerical simulations. Mathematically, the passage from kinetic to macroscopic models is performed by asymptotic analysis. From a numerical point of view, considering a small mean free path, the kinetic equation then contains stiff terms which make the numerical simulations very expensive for stability reasons. In fact, a typical example is the presence of multiple spatial and temporal scales which intervene in different positions and at different times. These behaviors make the construction of efficient numerical methods a real challenge.
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SIAM J. Sci. Comput., Sociey for Industrial and Applied Mathematics, 2016, 38 (2), pp.737-764
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  • HAL Id : hal-01130002, version 1
  • ARXIV : 1503.04586

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Nicolas Crouseilles, Hélène Hivert, Mohammed Lemou. Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium. SIAM J. Sci. Comput., Sociey for Industrial and Applied Mathematics, 2016, 38 (2), pp.737-764. 〈hal-01130002〉

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