Hi, I am trying to do a simple numerical calculation of diffusion equation. ∂t u(x, t) = ∂x^2 u(x, t). I could replicate a seemingly appropriate result ( ). However, I found the spectrum of the result is quite strange ( ). Since "∂x^2 u(x, t)" becomes "-k^2 uk" in fourier space (uk is fourier coefficient of u), the spectrum of large wave number region must be smaller. To confirm it, I solved the same equation using spectral method. d/dt uk = -k^2 uk. Then I obtained the third figure ( ) which seems appropriate. I interpret such a difference between finite difference and spectral methods in large wave number region as following. The finite difference equation is, ∂t u = [u(x+Δx) - 2u(x) + u(x-Δx)]/Δx^2. If we substitute u(x) = uk exp(ikx), the above equation becomes d/dt uk = uk [exp(ikΔx) + exp(-ikΔx) -2]/Δx^2 ≈ uk [-k^2 + O(k^4 Δx^2)]. Therefore, there remains O(k^4 Δx^2) difference with spectral method. Even if Δx is small enough, the large k region have big error. Now, could you tell me whether this interpretation is correct. And I would like to know the remedy of this problem.