The Unstable Mode in the Crank-Nicolson Leap-Frog Method Is Stable

Abstract

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.