Large wave number region of difference method of diffusion

[sanukisoba]

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.

 

[nilesh_pat]

Thank you.Yes, your interpretation is correct. The difference between finite difference and spectral methods in the large wave number region is due to the higher order terms that are neglected in the finite difference equation. To remedy this issue, you can use higher order finite difference schemes or you can use a filter to smooth out the high frequency components of the solution.