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.
Structure and asymptotic preserving deep neural surrogates for uncertainty quantification in multiscale kinetic equations
University of Ferrara（费拉拉大学）；Italian Ministry of University and Research（意大利大学与研究部）；Royal Society（皇家学会）；European Union（欧盟）；ICSC – Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing（ICSC – 高性能计算、大数据和量子计算研究中心）；Italian National Institute of High Mathematics（意大利国家高等数学研究所）
A Unified Gas-kinetic Particle Method for Multiscale Photon Transport
Beijing Computational Science Research Center（北京计算科学研究中心）；Hong Kong University of Science and Technology（香港科技大学）；Northwestern Polytechnical University（西北工业大学）
A Multiscale Kinetic-Fluid Solver with Dynamic Localization of Kinetic Effects
Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心图卢兹数学研究所）；Institut de Mathématiques de Toulouse UMR 5219（法国原子能和替代能源委员会萨克雷中心）；Commissariat à l’Energie Atomique CEA-Saclay DM2S-SFME（波尔多大学数学研究所）；Université de Bordeaux；Institut de Mathématiques
Explicit form of simplified Grad's 13 moments distribution function-based moment gas kinetic solver with unstructured meshes for the multiscale rarefied flow
An implicit multiscale solver for steady kinetic equations based on wave-particle decomposition
Hong Kong University of Science and Technology（香港科技大学）；Shenzhen Research Institute, Hong Kong University of Science and Technology（香港科技大学深圳研究院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）
