This report proves that under the time step condition $\bigtriangleup t|\Lambda|<1$(| $\cdot$ | = Euclidean
norm) suggested by root condition analysis and necessary for stability, all modes of the Crank-Nicolson Leap-Frog (CNLF) approximate solution to the system

$\frac{du}{dt}+ Au + \Lambda u = 0$, for $t > 0$ and $u(0) = u_0$,

where $A + A^T$ is symmetric positive definite and $\Lambda$ is skew symmetric, are asymptotically stable.
This result gives a sufficient stability condition for non-commutative $A$ and $\Lambda$, and is proven by
energy methods. Thus, the growth, often reported in the unstable mode, is not due to systems
effects and its explanation must be sought elsewhere.