Non-oscillatory spectral analysis: Elimination of Gibbs phenomena
from reconstructed piecewise continuous signals
Firmin Ndeges
Firmin Ndeges, Non-oscillatory spectral analysis: Elimination of
Gibbs phenomena from reconstructed piecewise continuous signals
Gibbs' phenomena occurs in Fourier approximation of piecewise
continuous functions with discontinuities. This can lead to a number
of practical difficulties including poor convergence behavior in
approximations to nonlinear partial differential equations. A remedy
for Gibbs phenomena was posed by Cai, Gottlieb, and Shu (Essentially
Non-Oscillatory Spectral Fourier Methods for Shock Wave
Calculations). In their paper, they show that adding a discontinuous
function to the standard trigonometric basis functions can mitigate
the highly oscillatory phenomena that persists even as more terms are
taken.
In this work, we validate this method for the case of one
discontinuity. We then extend the method for the case of multiple
discontinuities. We begin by detailing an algorithm for selecting
the appropriate basis functions to add to the (finite) trigonometric
set using only Fourier coefficients of the discontinuous function.
We then numerically validate our results for the case of two
discontinuities, and derive a scheme for the case of three
discontinuities or higher. Mathematical analysis is included that
verifies our computed results.
We finally look at more practical issues, such as the sensitivity of
our procedure to the magnitude and spacing of discontinuities. This
study contains both numerical experimentation and supporting
mathematical analysis. (advisors Borggaard, Baumann and Herdman)