[kaya]

I am looking at 2nd and 4th order central finite difference schemes and trying to understand how good they can resolve the first derivative for each given wave number.

My routine is I make a sine curve with given wavenumber k, take the derivative with given FD scheme and transfer the result into fourier space to compare the amplitude of the corresponding wave's exact derivative.

I also check 2nd and 4th order cell centered schemes where I sample my sine curve on a finer grid with half grid spacing and get derivatives in a staggered like manner. I expect these to have better resolution characteristics, which seems true for 4th order case but not for 2nd order case. It looks exactly the same as non-staggered 2nd order FD scheme, which I can't understand. To me I just halved the grid spacing so I expect the scheme to behave twice as good in terms of wave resolution.

I am putting the figure below where x axis is wave number and y is just normalized amplitude.cc- schemes are cell-centered / staggered. My nyquist wave number is 16. I believe that I am doing something wrong but wanted to hear some opinions...

Thanks