Homework Help: Large wave number region of difference method of diffusion

1. Jan 30, 2012

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.

Last edited: Jan 30, 2012