An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit

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• 作者：Li Wang^a ; ^{lwang46@buffalo.edu" class="auth_mail" title="E-mail the corresponding author} ; Bokai Yan^b ; ^{byan@math.ucla.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional diffusion ; Heavy tail ; Asymptotic-preserving scheme ; Micro&ndash ; macro decomposition
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：312
• 期：Complete
• 页码：157-174
• 全文大小：580 K

文摘

We present a new asymptotic-preserving scheme for the linear Boltzmann equation which, under appropriate scaling, leads to a fractional diffusion limit. Our scheme rests on novel micro–macro decomposition to the distribution function, which splits the original kinetic equation following a reshuffled Hilbert expansion. As opposed to classical diffusion limit, a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter. Our idea is, while solving the macro–micro part in a truncated velocity domain (truncation only depends on numerical accuracy), to incorporate an integrated tail over the velocity space that is beyond the truncation, and its major component can be precomputed once with any accuracy. Such an addition is essential to drive the solution to the correct asymptotic limit. Numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes.