# Von Nuemann Stability Analysis CN scheme

1. Feb 22, 2012

### edge333

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

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.