Recovering exponential accuracy in Fourier spectral methods involving piecewise smooth functions with unbounded derivative singularities

Zheng Chen Brown University Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1610.25027

Journal of Scientific Computing, 65, 1145-1165, 2015
Fourier spectral methods achieve exponential accuracy both on the approximation level and for solving partial differential equations (PDEs), if the solution is analytic. If the solution is discontinuous but piecewise analytic up to the discontinuities, Fourier spectral methods produce poor pointwise accuracy, but still maintains exponential accuracy after post-processing. In earlier work, an extended technique is provided to recover exponential accuracy for functions which have end-point singularities, from the knowledge of point values on standard collocation points. In this paper, we develop a technique to recover exponential accuracy from the first $N$ Fourier coefficients of functions which are analytic in the open interval but have unbounded derivative singularities at end points. With this post-processing method, we are able to obtain exponential accuracy of spectral methods applied to linear transport equations involving such functions.
Spectral method; exponential accuracy; Fourier coefficients; Gegenbauer expansion; transport equation; variable coefficients; singular initial conditions; noise
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@inproceedings{zheng2015recovering,
  title={ Recovering exponential accuracy in Fourier spectral methods involving piecewise smooth functions with unbounded derivative singularities},
  author={Zheng Chen, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012032042862364145},
  booktitle={Journal of Scientific Computing},
  volume={65},
  pages={1145-1165},
  year={2015},
}
Zheng Chen, and Chi-Wang Shu. Recovering exponential accuracy in Fourier spectral methods involving piecewise smooth functions with unbounded derivative singularities. 2015. Vol. 65. In Journal of Scientific Computing. pp.1145-1165. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012032042862364145.
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