Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit

TL;DR

This paper develops two asymptotic preserving numerical schemes for kinetic equations with anomalous diffusion scaling, effectively capturing fractional diffusion limits and maintaining accuracy across different regimes.

Contribution

It introduces two novel AP schemes tailored for anomalous diffusion in kinetic equations, with one achieving uniform accuracy across all scaling parameters.

Findings

Both schemes are AP and consistent with the kinetic and fractional diffusion models.

The second scheme is uniformly accurate with respect to the scaling parameter.

Numerical tests demonstrate the schemes' efficiency and robustness.

Abstract

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.