I'm recently studying for the spectral solver to implement EM-PIC code. I read an article and have some questions.
Many PIC codes uses spectral solver to overcome numerical artifacts on FDTD.
In the high-order approximation of spatial derivative in PSTD and PSATD scheme, $$ \partial_{x}E_{x}|^{n}_{i,j,k} = \Sigma_{l=0}^{p/2-1} c_{l,p}\frac{E_x|^{n}_{i+1/2+l,j,k}-E_x|^{n}_{i-1/2-l,j,k}}{\Delta x} $$ after Fourier transform due to the efficiency of calculation, $$ F[\partial_{x}E_{x}] = i[k_x]_p \hat{E_x} \\ [k_x]_p = \Sigma_{l=0}^{p/2-1} c_{l,p}\frac{e^{ik_x(l+1/2)\Delta x}-e^{-ik_x(l+1/2)\Delta x}}{\Delta x} $$ where $F$ represent the Fourier transform and $\hat{E_x} = F[E_x]$.
The term $[k_x]_p$ goes to $k_x$ as p goes to infinity. But commonly use values for $p$ in practice 32 or 64 (as the article said).
Is there any reason to use the $p$ value of 32 or 64 in practice?
If I let $p=\infty$, I don't need to use the above high-order approximation formula. Then, it becomes simpler and the computational cost will be reduced I think.
