Exclusive robustness of Gegenbauer method to truncated convolution errors

TL;DR

This paper demonstrates that the Gegenbauer spectral reconstruction method is uniquely robust against truncated convolution errors caused by discontinuities, maintaining high accuracy even when other methods fail, with implications for spectral methods in non-smooth problems.

Contribution

The paper proves and illustrates that the Gegenbauer method remains exceptionally robust against truncated convolution errors, unlike other spectral reconstruction techniques.

Findings

Gegenbauer method error diminishes as O(N^{-1}) with Fourier order N.

Gegenbauer method achieves exponential convergence regardless of constants.

Truncated convolution errors can be practically resolved in grating mode analysis.

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.