Why is FFT of (A+B) different from FFT(A) + FFT(B)?

Question

I have been fighting with a very weird bug for almost a month. Asking you guys is my last hope. I wrote a program in C that integrates the 2d Cahn–Hilliard equation using the Im

Answer 1

I solved it!! The problem was the calculation of the Nonl term:

  Nonl[i*Ny+j][0] = dh[i*Ny+j][0]*dh[i*Ny+j][0]*dh[i*Ny+j][0];
  Nonl[i*Ny+j][1] = 0.0;

That needs to be changed to:

  Nonl[i*Ny+j][0] = dh[i*Ny+j][0]*dh[i*Ny+j][0]*dh[i*Ny+j][0] -3.0*dh[i*Ny+j][0]*dh[i*Ny+j][1]*dh[i*Ny+j][1];
  Nonl[i*Ny+j][1] = -dh[i*Ny+j][1]*dh[i*Ny+j][1]*dh[i*Ny+j][1] +3.0*dh[i*Ny+j][0]*dh[i*Ny+j][0]*dh[i*Ny+j][1];

In other words: I need to consider dh as a complex function (even though it should be real).

Basically, because of stupid rounding errors, the IFT of the FT of a real function (in my case dh), is NOT purely real, but will have a very small imaginary part. By setting Nonl[i*Ny+j][1] = 0.0 I was completely ignoring this imaginary part. The issue, then, was that I was recursively summing FT(dh), dhft, and an object obtained using the IFT(FT(dh)), this is Nonlft, but ignoring the residual imaginary parts!

Nonlft[i*Ny+j][0] = -Q2[i*Ny+j]*(Nonlft[i*Ny+j][0] -dhft[i*Ny+j][0]);
Nonlft[i*Ny+j][1] = -Q2[i*Ny+j]*(Nonlft[i*Ny+j][1] -dhft[i*Ny+j][1]);

Obviously, calculating Nonlft as dh^3 -dh and then doing

Nonlft[i*Ny+j][0] = -Q2[i*Ny+j]* Nonlft[i*Ny+j][0]; 
Nonlft[i*Ny+j][1] = -Q2[i*Ny+j]* Nonlft[i*Ny+j][1];

Avoided the problem of doing this "mixed" sum.

Phew... such a relief! I wish I could assign the bounty to myself! :P

EDIT: I'd like to add that, before using the fftw_plan_dft_2d functions, I was using fftw_plan_dft_r2c_2d and fftw_plan_dft_c2r_2d (real-to-complex and complex-to-real), and I was seeing the same bug. However, I suppose that I couldn't have solved it if I didn't switch to fftw_plan_dft_2d, since the c2r function automatically "chops off" the residual imaginary part coming from the IFT. If this is the case and I'm not missing something, I think that this should be written somewhere on the FFTW website, to prevent users from running into problems like this. Something like "r2c and c2r transforms are not good to implement pseudospectral methods".

EDIT: I found another SO question that addresses exactly the same problem.