Finite difference weights, spectral differentiation, and superconvergence

Abstract:

Let $z_{1},z_{2},\ldots ,z_{N}$ be a sequence of distinct grid points. A finite difference formula approximates the $m$-th derivative $f^{(m)}(0)$ as $\sum w_{k}f\left (z_{k}\right )$, with $w_{k}$ being the weight at $z_{k}$. We derive an algorithm for finding the weights $w_{k}$ which uses fewer arithmetic operations and less memory than the algorithm in current use (Fornberg, Mathematics of Computation, vol. 51 (1988), pp. 699-706). The algorithm we derive uses fewer arithmetic operations by a factor of $(5m+5)/4$ in the limit of large $N$. The optimized C++ implementation we describe is a hundred to five hundred times faster than MATLAB. The method of Fornberg is faster by a factor of five in MATLAB, however, and thus remains the most attractive option for MATLAB users.

The algorithm generalizes easily to the calculation of spectral differentiation matrices, or equivalently, finite difference weights at several different points with a fixed grid. Unlike the algorithm in current use for the calculation of spectral differentiation matrices, the algorithm we derive suffers from no numerical instability.

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 $\mathcal {O}\left (h^{N-m}\right )$. However, the most commonly used finite difference formulas have an order of accuracy that is higher than normal. For instance, the centered difference approximation $\left (f(h)\!-\!2f(0)\!+\!f(-h)\right )$ $/h^{2}$ 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 $1$.