2015/05/13 by Nicolas Crouseilles, Crouseilles, Nicolas, Hélène Hivert +3
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Mathematical Biology Tumor Growth #Numerical Analysis (math.NA) #nanoparticles nucleation surface interactions
paper · doi:10.48550/arxiv.1505.03250
openalex publication_date 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.