论文标题
在各向同性中心差的概括中，分数laplacian的高阶近似值
On Generalisation of Isotropic Central Difference for Higher Order Approximation of Fractional Laplacian
论文作者
论文摘要
本文讨论了将整数秩序拉普拉斯对分数顺序概述的核心差异的研究。分析表明，与先前的研究的结论相反，通过快速傅立叶变换评估的差异模板可防止分数拉普拉斯溶液的收敛。我们提出了一个复合正交规则，以便用所需的收敛速率有效评估模板系数，以确保溶液的收敛性。此外，我们建议使用通用的高阶晶格Boltzmann方法来生成模板，该模板可以近似于较高级收敛速度和误差各向同性的分数laplacian。我们还回顾了晶格Boltzmann方法的配方，并讨论了使用Smolyak算法配制的显式稀疏溶液，以及评估Hermite多项式的方法，以有效地生成高阶模板。进行数值实验以验证误差分析和制剂。
The study of generalising the central difference for integer order Laplacian to fractional order is discussed in this paper. Analysis shows that, in contrary to the conclusion of a previous study, difference stencils evaluated through fast Fourier transform prevents the convergence of the solution of fractional Laplacian. We propose a composite quadrature rule in order to efficiently evaluate the stencil coefficients with the required convergence rate in order to guarantee convergence of the solution. Furthermore, we propose the use of generalised higher order lattice Boltzmann method to generate stencils which can approximate fractional Laplacian with higher order convergence speed and error isotropy. We also review the formulation of the lattice Boltzmann method and discuss the explicit sparse solution formulated using Smolyak's algorithm, as well as the method for the evaluation of the Hermite polynomials for efficient generation of the higher order stencils. Numerical experiments are carried out to verify the error analysis and formulations.
