Convergence of spectral methods for nonlinear conservation laws
SIAM Journal on Numerical Analysis
On one-sided filters for spectral Fourier approximations of discontinuous functions
SIAM Journal on Numerical Analysis
Family of spectral filters for discontinuous problems
Journal of Scientific Computing
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Legendre pseudospectral viscosity method for nonlinear conservation laws
SIAM Journal on Numerical Analysis
Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions
Mathematics of Computation
A note on the accuracy of spectral method applied to nonlinear conservation laws
Journal of Scientific Computing
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Determination of the jumps of a bounded function by its Fourier series
Journal of Approximation Theory
On a high order numerical method for functions with singularities
Mathematics of Computation
Enhanced spectral viscosity approximations for conservation laws
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
Detection of Edges in Spectral Data II. Nonlinear Enhancement
SIAM Journal on Numerical Analysis
EXPONENTIALLY ACCURATE APPROXIMATIONS TO PIECE-WISE SMOOTH PERIODIC FUNCTIONS
EXPONENTIALLY ACCURATE APPROXIMATIONS TO PIECE-WISE SMOOTH PERIODIC FUNCTIONS
Chebyshev super spectral viscosity method for a fluidized bed model
Journal of Computational Physics
On inverse methods for the resolution of the Gibbs phenomenon
Journal of Computational and Applied Mathematics
Edge Detection Free Postprocessing for Pseudospectral Approximations
Journal of Scientific Computing
Iterative methods based on spline approximations to detect discontinuities from Fourier data
Journal of Computational and Applied Mathematics
Journal of Computational Physics
Gibbs phenomenon removal by adding Heaviside functions
Advances in Computational Mathematics
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Consider a piecewise smooth function for which the (pseudo-)spectral coefficients are given. It is well known that while spectral partial sums yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Various types of reconstruction methods developed for the recovery of piecewise smooth functions have met with varying degrees of success. The Gegenbauer reconstruction method, originally proposed by Gottlieb et al. has the particularly impressive ability to reconstruct piecewise analytic functions with exponential convergence up to the points of discontinuity. However, it has been sharply criticized for its high cost and susceptibility to round-off error. In this paper, a new approach to Gegenbauer reconstruction is considered, resulting in a reconstruction method that is less computationally intensive and costly, yet still enjoys superior convergence. The idea is to create a procedure that combines the well known exponential filtering method in smooth regions away from the discontinuities with the Gegenbauer reconstruction method in regions close to the discontinuities. This hybrid approach benefits from both the simplicity of exponential filtering and the high resolution properties of the Gegenbauer reconstruction method. Additionally, a new way of computing the Gegenbauer coefficients from Jacobian polynomial expansions is introduced that is both more cost effective and less prone to round-off errors.