Stable calculation of Gaussian-based RBF-FD stencils
Stable calculation of Gaussian-based RBF-FD stencils
复制标题
DOI：
10.1016/j.camwa.2012.11.006
复制
发表时间：
2013-02-01
影响因子：
2.9
通讯作者：
Powell, Collin
中科院分区：
文献类型：
--
作者：
Fornberg, Bengt;Lehto, Erik;Powell, Collin
传统的有限差分方法被设计成对于低次多项式是精确的。它们在Cartesian类型的网格上非常有效，但对于非结构化的节点布局可能会失败。径向基函数生成的有限差分（RBF-FD）方法克服了这个问题，因此，提供了一个更好的几何灵活性。RBF-FD权重的计算涉及形状参数θ。小的RBF值（对应于接近平坦的RBFs）通常会导致特别准确的RBF-FD公式。然而，最直接的方法来计算权重（径向基函数直接），然后成为数字高度病态。相比之下，本算法在进入epsilon -> 0极限时一直保持数值稳定。与RBF-QR算法一样，它使用在与病态近似平坦的原始高斯径向基函数所跨越的相同函数空间中找到数值上良好条件的基函数集的思想。利用不完全伽玛函数的一些性质，我们发现基的变换可以在不进行无限展开的情况下实现。与Contour-Pade算法、RBF-RA算法和RBF-QR算法进行了比较，分析了其优缺点。(C)2012爱思唯尔有限公司保留所有权利。
Traditional finite difference (FD) methods are designed to be exact for low degree polynomials. They can be highly effective on Cartesian-type grids, but may fail for unstructured node layouts. Radial basis function-generated finite difference (RBF-FD) methods overcome this problem and, as a result, provide a much improved geometric flexibility. The calculation of RBF-FD weights involves a shape parameter epsilon. Small values of epsilon (corresponding to near-flat RBFs) often lead to particularly accurate RBF-FD formulas. However, the most straightforward way to calculate the weights (RBF-Direct) becomes then numerically highly ill-conditioned. In contrast, the present algorithm remains numerically stable all the way into the epsilon -> 0 limit. Like the RBF-QR algorithm, it uses the idea of finding a numerically well-conditioned basis function set in the same function space as is spanned by the ill-conditioned near-flat original Gaussian RBFs. By exploiting some properties of the incomplete gamma function, it transpires that the change of basis can be achieved without dealing with any infinite expansions. Its strengths and weaknesses compared with the Contour-Pade, RBF-RA, and RBF-QR algorithms are discussed. (C) 2012 Elsevier Ltd. All rights reserved.