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Large wave number region of difference method of diffusion

  1. Jan 30, 2012 #1
    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 ( 1.png ).

    However, I found the spectrum of the result is quite strange ( 2.png ).
    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 ( 3.png ) 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
  2. jcsd
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