Comparing RBF‐FD approximations based on stabilized Gaussians and on polyharmonic splines with polynomials
Comparing RBF‐FD approximations based on stabilized Gaussians and on polyharmonic splines with polynomials
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比较基于稳定高斯和多调和样条的 RBF-FD 近似与多项式
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Emerson Abreu
中科院分区：
文献类型：
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作者：
L. G. C. Santos;N. Manzanares;G. Menon;Emerson Abreu
在这项工作中，我们关注径向基函数生成的有限差分（RBF-FD）近似。使用平方域中泊松方程的一些解析解，提出了稳定平坦高斯函数 (RBF(SGA)-FD) 和具有补充多项式的多调和样条函数 (RBF(PHS)-FD) 的数值误差估计。结构化和非结构化点云都用于评估 RBF(PHS)-FD 中的云细化、局部支撑尺寸和多项式的最大允许次数的影响。 RBF(SGA)-FD 和 RBF(PHS)-FD 均获得了高精度，特别是对于非结构化云。还使用其中一个解析解在域的所有点估计一阶和二阶导数的绝对误差。对于 RBF(SGA)-FD，该测试显示了边界邻域上一些局部偏心支撑的不当情况。 RBF(PHS)-FD 没有观察到这种现象。
In this work, we are concerned with radial basis function–generated finite difference (RBF‐FD) approximations. Numerical error estimates are presented for stabilized flat Gaussians (RBF(SGA)‐FD) and polyharmonic splines with supplementary polynomials (RBF(PHS)‐FD) using some analytical solutions of the Poisson equation in a square domain. Both structured and unstructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports, and maximal permissible degree of the polynomials in RBF(PHS)‐FD. High order of accuracy was attained with both RBF(SGA)‐FD and RBF(PHS)‐FD especially for unstructured clouds. Absolute errors in the first and second derivatives were also estimated at all points of the domain using one of the analytical solutions. For RBF(SGA)‐FD, this test showed the occurrence of improprieties of some decentered supports localized on boundary neighborhoods. This phenomenon was not observed with RBF(PHS)‐FD.