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)