How Can Numerical Stability Be Achieved in Unsteady Laminar Flow Equations?

[Aero51]

I took a CFD class last semester (had to leave school though due to personal garbage). I am making a come back this fall and as some extra credit I am trying to numerically solve the unsteady laminar flow equation in a pipe. The equation is

\dot{U} + U'' + K = 0
where dots denote the time derivative and primes denote spatial derivatives (in this case the radius, r)

The discretization of the equation is:
(U^{n+1}_{i}-U^{n}_{i})/\Delta T + (U^{n}_{i+1} - 2U^{n}_i+U^n_{i-1})/(\Delta R)^2 + K

However, when I try to do the stability analysis I get this really ugly problem:
CFL = 1-2{\Delta T}/(\Delta R)^2 * (cosh(\Delta R)-1)-K\Delta T e^{i K_m R}

Any ideas how to eliminate that last euler number to make the stability analysis more feasible?

 

[Aero51]

Update: I am going to try to take the natural log and multiply by its conjugate base and see what happens

 

[verty]

Since you are already retaking this class, I would just look up the answer. You have tried your hardest to solve it, now find the answer. It just isn't worth the energy to get stuck on something like this.

PS. And of course I mean, find the answer and try to understand it, read more online, etc.

 

[Aero51]

I talked it over with my professor, he actually wasn't sure how to deal with the constant term either. However, he did mention that he was certain the stability analysis was only dependent on the unsteady and viscus terms. Also, I had a coefficient negative when it should have been positive...