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Exclusive robustness of Gegenbauer method to truncated convolution errors

Faghihifar, E ; Sharif University of Technology | 2022

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  1. Type of Document: Article
  2. DOI: 10.1016/j.jcp.2021.110911
  3. Publisher: Academic Press Inc , 2022
  4. Abstract:
  5. 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. These errors 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. fˆk≈∑|ℓ|⩽Ngˆℓhˆk−ℓ. 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 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. © 2021 Elsevier Inc
  6. Keywords:
  7. Gegenbauer method ; Gibbs phenomenon ; Grating modes ; Differential equations ; Errors ; Fourier analysis ; Fourier series ; Modal analysis ; Gegenbaue method ; Gegenbauer ; Gibbs phenomena ; Non-smooth differential equations ; Performance ; Reconstruction techniques ; Spectral reconstruction ; Spectral solutions ; Truncated convolution error ; Convolution
  8. Source: Journal of Computational Physics ; Volume 452 , 2022 ; 00219991 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S0021999121008068