On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries
On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries
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DOI：
10.1016/j.jcp.2018.12.013
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发表时间：
2019-03-01
影响因子：
4.1
通讯作者：
Fornberg, Bengt
中科院分区：
文献类型：
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作者：
Bayona, Victor;Flyer, Natasha;Fornberg, Bengt
径向基函数生成有限差分（RBF-FD）近似将基于网格的正则有限差分推广到离散节点集。当它们基于多变量多项式（PHS+poly）增强的多调和样条（PHS）时，这些变得特别有效。一个关键的特点是，可以实现高精度的订单，而不必选择一个最佳的形状参数，而不必处理与数值病态的问题。这种方法的优势，以前被证明是特别引人注目的近似域边界附近，在那里的支架成为高度片面的。由于多项式龙格现象，在这种情况下，高精度的常规FD近似将具有非常大的权重。在生成权重的过程中包括PHS型RBF使得可以避免这种不利影响。以此为动机，本研究旨在更好地理解PHS+多生成RBF-FD近似在边界附近的行为，并在1-D，2-D和3-D中对其进行说明。(C)2019爱思唯尔公司All rights reserved.
Radial basis function generated finite difference (RBF-FD) approximations generalize grid-based regular finite differences to scattered node sets. These become particularly effective when they are based on polyharmonic splines (PHS) augmented with multi-variate polynomials (PHS+poly). One key feature is that high orders of accuracy can be achieved without having to choose an optimal shape parameter and without having to deal with issues related to numerical ill-conditioning. The strengths of this approach were previously shown to be especially striking for approximations near domain boundaries, where the stencils become highly one-sided. Due to the polynomial Runge phenomenon, regular FD approximations of high accuracy will in such cases have very large weights well into the domain. The inclusion of PHS-type RBFs in the process of generating weights makes it possible to avoid this adverse effect. With that as motivation, this study aims at gaining a better understanding of the behavior of PHS+poly generated RBF-FD approximations near boundaries, illustrating it in 1-D, 2-D and 3-D. (C) 2019 Elsevier Inc. All rights reserved.