分数阶Laplacian高阶逼近的迷向中心差分的推广
On Generalisation of Isotropic Central Difference for Higher Order Approximation of Fractional Laplacian
摘要：
本文讨论了将整数阶拉普拉斯算子的中心差分推广到分数阶的问题。分析表明，与以往研究的结论相反，通过快速傅里叶变换求出的差分展度阻止了分数阶拉普拉斯算子解的收敛。我们提出了一个复合求积规则，以有效地评估模板系数所需的收敛速度，以保证收敛的解决方案。此外，我们提出了使用广义高阶格子玻尔兹曼方法来生成可以近似分数拉普拉斯算子的高阶收敛速度和误差各向同性的张量。我们还回顾了制定的晶格玻尔兹曼方法，并讨论了显式稀疏的解决方案制定使用Schwarak的算法，以及方法的评价的Hermite多项式的有效生成的高阶stenosis。数值实验验证了误差分析和公式。
英文摘要：
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.
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