We propose a novel method to compute a finite difference stencil for Riesz derivative for artibitrary speed of convergence. This method is based on applying a pre-filter to the Gr\"unwald-Letnikov type central difference stencil. The filter is obtained by solving for the inverse of a symmetric Vandemonde matrix and exploiting the relationship between the Taylor's series coefficients and fast Fourier transform. The filter costs O\left(N^{2}\right) operations to evaluate for O\left(h^{N}\right) of convergence, where h is the sampling distance. The higher convergence speed should more than offset the overhead with the requirement of the number of nodal points for a desired error tolerance significantly reduced. The benefit of progressive generation of the stencil coefficients for adaptive grid size for dynamic problems with the Gr\"unwald-Letnikov type difference scheme is also kept because of the application of filtering. The higher convergence rate is verified through numerical experiments.

翻译：我们建议一种新颖的方法来计算Riesz衍生物的有限差异, 以精度趋同速度 。 这种方法的基础是将预过滤器应用到 Gr\"unwald- Letnikov 类型中心差异 stencils。 过滤器的获取方法是解决对称 Vandemonde 矩阵的反向, 利用泰勒的序列系数和快速 Fourier 变异之间的关系。 过滤器成本 O\left (N ⁇ 2 ⁇ right) 操作来评价 O\left (h ⁇ N ⁇ right) 趋同( h) 是取样距离 。 更高的趋同速度应该超过对预期误差容忍的节点数要求来抵消间接费用。 渐进生成用于适应与 Gr\ " unwald- Letnikov 类型差异的网格大小的Stenci 系数的好处也由于过滤的应用而得以保留。 通过数字实验来验证更高的趋同率 。