Pseudo-spectral methods

The idea

A periodic function is a sum of sine waves. Differentiating a sine wave just multiplies it by its wavenumber. So if you store a field as its Fourier coefficients $\hat u_k$, every spatial derivative is a multiplication:

$$\widehat{\partial_x u}_k = i k\, \hat u_k, \qquad \widehat{\partial_x^3 u}_k = (ik)^3\, \hat u_k .$$

No finite-difference stencil, no truncation error in the derivative. For smooth fields the error decays faster than any power of the grid spacing. This is called spectral accuracy.

The “pseudo” part

Nonlinear terms like $u^2$ are awkward in Fourier space (they become convolutions). So the engine goes back and forth:

1. inverse FFT to get $u$ on the grid,
2. compute $u^2$ pointwise,
3. FFT back and multiply by $-3ik$.

That round trip is the pseudo-spectral method. The FFT makes it cost $O(N \log N)$ per evaluation.

Grid and wavenumbers

The engine uses a periodic grid on $[-L/2, L/2)$ with $N$ points, and wavenumbers in FFT order $k = \frac{2\pi}{L}\,(0, 1, \dots, N/2 - 1, -N/2, \dots, -1)$. Pick $L$ large enough that nothing important reaches the boundary, and $N$ large enough that the Fourier spectrum of the solution has decayed to round-off before the highest wavenumber.

What is not yet here

• Dealiasing (the 2/3 rule) for strongly nonlinear or under-resolved runs.
• Non-periodic boundaries via Chebyshev or mapped domains.

Both are on the roadmap.