Could spectral methods solve first-order differential equations?

Question

I've been learning spectral methods mainly through reading the book "spectral mehtods in matlab" by Trefethen. I have a simple question: Could spectral methods solve first-order differential equations? It seems using spectral methods requires two initial conditions.

Answer 1

The spectral method uses Fourier transforms or the Chebyshevs differentiation matrix to convert a space-time PDE to a time-dependent ODE. The way it does it is by approximating the spatial derivatives with high precision and then usually finite differences or the Runge-Kutta method is used to solve the temporal ODEs.

You can use the spectral method to solve ODEs also. But using the Runge-Kutta method would usually be a much better choice.

If you are solving a first-order PDE, then you should use the spectral method to convert your PDE to a time-dependent ODE, and then use finite differences or the RK method to solve that ODE.