when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability:

One factor would be the condition number of the approximation operator. The other factor would be the CFL Condition. I think I understand both, but :

1) How come CFL condition only depends on the equation , while the Condition number is dependent on the approximation operator as well?

2) Are the two dependent? Can one be deduced from the other?

The problem in which I've encountered the dilemma is the one dimensional Biharmonic Linear Schrodinger equation, but I think the aforementioned is far more general then a specific problem.

$iu_t = u_{xxxx}$, $x$ is periodic in $[0, 2 \pi]$, $t>0$ and $u(x,0) = u_0 (x)\in H^2$