Comparative numerical laboratory
Spectral Methods: Chebyshev Accuracy and Gibbs Behavior — Chapter 13
Compare smooth spectral convergence with the oscillatory behavior near a discontinuity.
\[
x_j=\cos\!\left(\frac{j\pi}{N}\right), j=0,\ldots,N.
\]
Methods compared
| Method | Convenience | Inconvenience / warning |
|---|---|---|
| Chebyshev interpolation | Fast convergence for smooth functions. | Global method; sensitive to nonsmoothness. |
| Fourier partial sums | Natural for periodic data. | Gibbs oscillations near jumps. |
| Filtering/dealiasing | Controls high-mode pollution. | May reduce resolution or alter sharp features. |
Illustrative plot / animation
Numerical diagnostics
What the student must compare
- coefficient decay
- endpoint clustering
- Gibbs oscillation
- smooth vs nonsmooth contrast