Hélène Hivert

Nicolas Crouseilles, Hélène Hivert, Mohammed Lemou

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.

Hélène Hivert

In this work, we propose an asymptotic preserving scheme for a non-linear kinetic reaction-transport equation, in the regime of sharp interface. With a non-linear reaction term of KPP-type, a phenomenon of front propagation has been proved in [9]. This behavior can be highlighted by considering a suitable hyperbolic limit of the kinetic equation, using a Hopf-Cole transform. It has been proved in [6, 8, 11] that the logarithm of the distribution function then converges to the viscosity solution of a constrained Hamilton-Jacobi equation. The hyperbolic scaling and the Hopf-Cole transform make the kinetic equation stiff. Thus, the numerical resolution of the problem is challenging, since the standard numerical methods usually lead to high computational costs in these regimes. The Asymptotic Preserving (AP) schemes have been typically introduced to deal with this difficulty, since they are designed to be stable along the transition to the macroscopic regime. The scheme we propose is adapted to the non-linearity of the problem, enjoys a discrete maximum principle and solves the limit equation in the sense of viscosity. It is based on a dedicated micro-macro decomposition, attached to the Hopf-Cole transform. As it is well adapted to the singular limit, our scheme is able to cope with singular behaviors in space (sharp interface), and possibly in velocity (concentration in the velocity distribution). Various numerical tests are proposed, to illustrate the properties and the efficiency of our scheme.

Nicolas Crouseilles, Hélène Hivert, Mohammed Lemou

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 \emph{Asymptotic Preserving} (AP): they are consistent with the kinetic equation for all fixed value of the scaling parameter $\varepsilon >0$ and degenerate into a consistent scheme solving the asymptotic model when $\varepsilon$ tends to $0$. The second scheme enjoys the stronger property of being uniformly accurate (UA) with respect to $\varepsilon$. 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.