Von Nuemann Stability Analysis CN scheme

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Homework Statement

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)

Homework Equations

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 = $\frac{1 - i 0.5 \frac{u Δt}{Δx} sin (β Δx) + \frac{D Δt}{Δx^{2}} (cos (β Δx) - 2) - 0.5k }{1+ i 0.5 \frac{u Δt}{Δx} sin (β Δx) - \frac{D Δt}{Δx^{2}} (cos (β Δx) - 2) - 0.5k }$

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.

 

[KataKoniK]

To eliminate the imaginary number term in the denominator, you can use the conjugate of the complex number in the denominator. The conjugate of a complex number a+bi is a-bi. In this case, you can multiply both the numerator and denominator by the complex conjugate of the denominator, which will result in the elimination of the imaginary number in the denominator. The resulting expression will still be complex, but you can then square it and take the absolute value to determine stability.