有限差分权重、谱微分与超收敛
Finite Difference Weights, Spectral Differentiation, and Superconvergence
AI总结：
提出一种改进的Fornberg算法，以更少的算术运算精确计算有限差分权重，并证明非对称公式的精度阶数最多可提升1。
摘要：
设$z_{1},z_{2},...,z_{N}$为一系列不同的网格点。有限差分公式将$m$阶导数$f^{(m)}(0)$近似为$\sum w_{k}f(z_{k})$，其中$w_{k}$为权重。我们推导了一种求解权重$w_{k}$的算法，该算法是对Fornberg算法（《计算数学》，第51卷（1988年），第699-706页）的改进。该算法使用的算术运算比Fornberg算法少$4/(5m+5)$倍，同时精度相同。我们推导的算法即使在导数阶数$m$高达16时也能精确计算有限差分权重。此外，该算法易于推广到谱微分矩阵的高效计算。 对于网格点$hz_{k}$（$1\leq k\leq N$），$f^{(m)}(0)$的有限差分公式的精度阶数通常为$\mathcal{O}(h^{N-m})$。然而，最常用的有限差分公式的精度阶数高于典型值。例如，中心差分近似$(f(h)-2f(0)+f(-h))/h^{2}$对$f''(0)$的精度阶数为2而非1。即使是非对称有限差分公式也可能表现出这种超收敛或提升的精度阶数，正如我们推导的显式代数条件所示。如果网格点是实数，我们证明了一个基本结果：精度阶数提升最多不超过1。
英文摘要：
Let $z_{1},z_{2},...,z_{N}$ be a sequence of distinct grid points. A finite difference formula approximates the $m$-th derivative $f^{(m)}(0)$ as $\sum w_{k}f(z_{k})$, with $w_{k}$ being the weights. We derive an algorithm for finding the weights $w_{k}$ which is an improvement of an algorithm of Fornberg (\emph{Mathematics of Computation}, vol. 51 (1988), p. 699-706). This algorithm uses fewer arithmetic operations than that of Fornberg by a factor of $4/(5m+5)$ while being equally accurate. The algorithm that we derive computes finite difference weights accurately even when $m$, the order of the derivative, is as high as 16. In addition, the algorithm generalizes easily to the efficient computation of spectral differentiation matrices. The order of accuracy of the finite difference formula for $f^{(m)}(0)$ with grid points $hz_{k}$, $1\leq k\leq N$, is typically $\mathcal{O}(h^{N-m})$. However, the most commonly used finite difference formulas have an order of accuracy that is higher than the typical. For instance, the centered difference approximation $(f(h)-2f(0)+f(-h))/h^{2}$ to $f"(0)$ has an order of accuracy equal to 2 not 1. Even unsymmetric finite difference formulas can exhibit such superconvergence or boosted order of accuracy, as shown by the explicit algebraic condition that we derive. If the grid points are real, we prove a basic result stating that the order of accuracy can never be boosted by more than 1.
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