具有分数阶扩散极限的Lévy-Fokker-Planck方程的渐近保持格式
An asymptotic preserving scheme for Lévy-Fokker-Planck equation with fractional diffusion limit
摘要：
本文发展了一种求解分数阶扩散标度Lévy-Fokker-Planck方程的数值方法。有两个主要挑战。一个来自双重非定域性，即需要将分数拉普拉斯算子应用于幂律衰减分布。另一个是由长时间/小平均自由程尺度引起的，这给方程引入了刚度。为了解决第一个困难，我们使用一个变量变换，将无界区域转化为有界区域，然后应用基于切比雪夫多项式的准谱方法。为了处理多尺度，我们提出了一个渐进保持计划的基础上，一个新的微观-宏观分解，使用的结构的检验函数在证明分数扩散极限解析。最后，通过一组数值算例说明了该格式的有效性和精度.
英文摘要：
In this paper, we develop a numerical method for the Lévy-Fokker-Planck equation with the fractional diffusive scaling. There are two main challenges. One comes from a two-fold nonlocality, that is, the need to apply the fractional Laplacian operator to a power law decay distribution. The other arises from long-time/small mean-free-path scaling, which introduces stiffness to the equation. To resolve the first difficulty, we use a change of variable to convert the unbounded domain into a bounded one and then apply the Chebyshev polynomial based pseudo-spectral method. To treat the multiple scales, we propose an asymptotic preserving scheme based on a novel micro-macro decomposition that uses the structure of the test function in proving the fractional diffusion limit analytically. Finally, the efficiency and accuracy of our scheme are illustrated by a suite of numerical examples.
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