---
tag: T37Q055GDA
kind: definition
name: Pseudo-spectral method
msc: [65M70, 65N35]
defines: XB91QS2AR0
uses: [MYV657YRDW, SVABG134WE, TST0GH66A5, S966EZX3ZT, M6EAR1JHPP]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudo-spectral_method
---

## Definition

Let $x_0, \ldots, x_{N-1}$ be collocation points in a domain $\Omega$, and let $V_N$ be a space of global functions with an interpolation property.
For every vector of values $(v_j)$ there is a unique $p \in V_N$ with $p(x_j) = v_j$ for all $j$.
Write $I_N v$ for this interpolant.
A *pseudo-spectral method* for an equation $u_t = \mathcal{N}(u)$, with $\mathcal{N}$ a differential operator, is the following scheme:

1. the unknowns are the values $u_j(t) \approx u(x_j, t)$;
2. each derivative $\partial^m u$ is replaced by $\partial^m (I_N u)$, evaluated at the points $x_j$;
3. products and other nonlinear functions of $u$ and its derivatives are evaluated pointwise at the $x_j$;
4. the equation is required to hold at each point $x_j$.

It is a [[spectral method|MYV657YRDW]] of collocation type.

## Remarks

Periodic case: $\Omega = [0, 2\pi)$, $x_j = 2\pi j / N$, and $V_N$ the [[trigonometric interpolants|SVABG134WE]] of degree about $N/2$.
Derivatives are then computed with the [[discrete Fourier transform|TST0GH66A5]]: transform, multiply the coefficient of $e^{ikx}$ by $ik$, and transform back.
With the [[fast Fourier transform|S966EZX3ZT]] this costs $O(N \log N)$ operations.
On an interval one often uses Chebyshev points $x_j = \cos(\pi j / (N - 1))$ and the [[Chebyshev polynomials|M6EAR1JHPP]].

Example: for $N = 8$ and $u_j = \sin x_j$, the transform has only the coefficients of $e^{\pm ix}$.
Multiplying them by $\pm i$ and transforming back gives $\cos x_j$ exactly.
For Burgers' equation $u_t + u u_x = 0$, one forms $u_x$ this way and then multiplies by $u_j$ at each point.

Non-example: a spectral Galerkin method computes the product $u u_x$ through its exact Fourier coefficients, a convolution of coefficient sequences.
It does not use pointwise evaluation, so clause 3 fails.