Exclusive robustness of Gegenbauer method to truncated convolution errors

Abstract

Spectral reconstructions provide rigorous means to remove the Gibbs phenomenon and accelerate the convergence of spectral solutions in non-smooth differential equations. In this paper, we show the concurrent emergence of truncated convolution errors could entirely disrupt the performance of most reconstruction techniques in the vicinity of discontinuities. They arise when the Fourier coefficients of the product of two discontinuous functions, namely $f=gh$, are approximated via truncated convolution of the corresponding Fourier series, i.e. $\hat{f}_k\approx \sum_{|\ell|\leqslant N}{\hat{g}_\ell\hat{h}_{k-\ell}}$. Nonetheless, we numerically illustrate and rigorously prove that the classical Gegenbauer method remains exceptionally robust against this phenomenon, with the reconstruction error still diminishing proportional to $\mathcal{O}(N^{-1})$ for the Fourier order $N$, and exponentially fast regardless of a constant. Finally, as a case study and a problem of interest in grating analysis whence the phenomenon initially was noticed, we demonstrate the emergence and practical resolution of truncated convolution errors in grating modes, which constitute the basis of Fourier modal methods.