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(adsbygoogle = window.adsbygoogle || []).push({});

[itex] \dot{U} + U'' + K = 0[/itex]

where dots denote the time derivative and primes denote spacial derivatives (in this case the radius, r)

The discretization of the equation is:

[itex](U^{n+1}_{i}-U^{n}_{i})/\Delta T + (U^{n}_{i+1} - 2U^{n}_i+U^n_{i-1})/(\Delta R)^2 + K[/itex]

However, when I try to do the stability analysis I get this really ugly problem:

[itex]CFL = 1-2{\Delta T}/(\Delta R)^2 * (cosh(\Delta R)-1)-K\Delta T e^{i K_m R}[/itex]

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

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# Numerical Stability of PDE

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