Recovering Exponential Accuracy in Fourier Spectral Methods Involving Piecewise Smooth Functions with Unbounded Derivative Singularities

Fourier spectral methods achieve exponential accuracy both on the approximation level and for solving partial differential equations (PDEs), if the solution is analytic. For linear PDEs with analytic but discontinuous solutions, Fourier spectral method produces poor pointwise accuracy, but still maintains exponential accuracy after post-processing [7]. 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.