I'm trying to understand pseudospectral methods in the context of solving PDEs. However, I can't seem to find a solid definition for this.
Is it simply a general term for solving a problem in parts: partly in the time domain and partly in the spectral domain (via the Fourier transform)?
Any good references or tips welcomed! Thanks.
Spectral methods look for approximations of solutions of an equation as a linear combination of a determined set of fucnctions, ussualy a a complete orthonormal system with respect to some weight. The firs step is to be able to approximate a function in that way . To fix things, fix an interval $[a,b]$ and a complete orthonormal system $\{\phi_k(x)\}_{k=1}^\infty$ with respect to a weighrt $w\colon[a,b]\to(0,\infty)$: $$ \int_a^b\phi_i(x)\,\phi_j(x)\,w(x)\,dx=\begin{cases}0 & \text{if }i\ne j,\\1 & \text{if }i=j.\end{cases} $$ For any $f\in L^2([a,b],w)$, that is, $\int_a^b|f|^2\,w\,dx<\infty$, we have $$ f=\sum_{k=1}^\infty\hat{f_k}\,\phi_k(x),\quad \hat{f_k}=\int_a^bf(x)\,\phi_k(x)\,w(x)\,dx, $$ with convergence in the $L^2$ sense. We approximate $f$ by the partial sum $$ f_N=\sum_{k=1}^N\hat{f_k}\,\phi_k(x). $$ This is an espectral approximations. In practice we may not know the function $f$ , but only its values at certain points $\{x_k\}_{k=1}^N$. Then we approximate $f$ by $$ \tilde{f_N}=\sum_{k=1}^N\tilde{f_k}\,\phi_k(x) $$ where the coefficients $\tilde{f_k}$ are chosen so that $f$ and $\tilde{f}$ coincide at the collocation points $\{x_k\}_{k=1}^N$. This is a pseudoespectral approximation.