STABLE COMPUTATION OF DIFFERENTIATION MATRICES AND SCATTERED NODE STENCILS BASED ON GAUSSIAN RADIAL BASIS FUNCTIONS
STABLE COMPUTATION OF DIFFERENTIATION MATRICES AND SCATTERED NODE STENCILS BASED ON GAUSSIAN RADIAL BASIS FUNCTIONS
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DOI：
10.1137/120899108
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发表时间：
2013-01-01
影响因子：
3.1
通讯作者：
Fornberg, Bengt
中科院分区：
文献类型：
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作者：
Larsson, Elisabeth;Lehto, Erik;Fornberg, Bengt
径向基函数（RBF）近似有可能提供光谱精确的函数近似的数据在分散的节点位置。对于光滑解，当基函数被缩放到接近平坦时，通常实现给定数目的节点的最佳精度。这也导致了几乎线性相关的基函数和严重的病态的插值矩阵。Fornberg，Larsson和Flyer最近推广了RBF-QR方法，为多达三维的任意节点集提供了一种数值稳定的插值方法，该方法具有平坦和近似平坦的高斯RBFs。在这项工作中，我们考虑如何将这种方法扩展到计算微分矩阵和模板权重的任务，以解决偏微分方程。建立了一阶、二阶导数算子和高粘性算子的表达式，解决了如何处理非均匀性等数值问题，并对该方法的精度和计算效率进行了数值检验.结果表明，使用RBF-QR方法求解偏微分方程问题可以很有竞争力相比，使用病态的直接解决方案的方法或使用可变精度算法，以克服条件的问题。
Radial basis function (RBF) approximation has the potential to provide spectrally accurate function approximations for data given at scattered node locations. For smooth solutions, the best accuracy for a given number of node points is typically achieved when the basis functions are scaled to be nearly flat. This also results in nearly linearly dependent basis functions and severe ill-conditioning of the interpolation matrices. Fornberg, Larsson, and Flyer recently generalized the RBF-QR method to provide a numerically stable approach to interpolation with flat and nearly flat Gaussian RBFs for arbitrary node sets in up to three dimensions. In this work, we consider how to extend this method to the task of computing differentiation matrices and stencil weights in order to solve partial differential equations. The expressions for first and second order derivative operators as well as hyperviscosity operators are established, numerical issues such as how to deal with non-unisolvency are resolved, and the accuracy and computational efficiency of the method are tested numerically. The results indicate that using the RBF-QR approach for solving PDE problems can be very competitive compared with using the ill-conditioned direct solution approach or using variable precision arithmetic to overcome the conditioning issue.