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Von Nuemann Stability Analysis CN scheme

  1. Feb 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Perform a von Nueman stability analysis for the Crank-Nicholson FTCS scheme for the following ADE:

    dc/dt + u dc/dx = D d2c/dx2 - kc

    where c is the concentration of interest, u is the advecting velocity and D is the dispersion coefficient.

    We used a general forward time, centered space numerical scheme to solve the above partial differential equation ( θ = 0.5 Crank-Nicholson)


    2. Relevant equations



    3. The attempt at a solution

    I got through most of the stability analysis including substituting in the errors for the finite difference approximation and solving for the value G:

    G = [itex]\frac{1 - i 0.5 \frac{u Δt}{Δx} sin (β Δx) + \frac{D Δt}{Δx^{2}} (cos (β Δx) - 2) - 0.5k }{1+ i 0.5 \frac{u Δt}{Δx} sin (β Δx) - \frac{D Δt}{Δx^{2}} (cos (β Δx) - 2) - 0.5k }[/itex]

    My question is, how do I eliminate the imaginary number term in the denominator, so that I may square G to determine whether it is less than 1 and thus stable? Do I use complex conjugates? If so, how? There are so many terms in the denominator that it just looks nightmare-ish.
     
  2. jcsd
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