A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions
Abstract
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 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.
Here, 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.
Several examples are discussed.
