THE UNSTABLE MODE IN THE CRANK-NICOLSON LEAP -FROG METHOD IS STABLE

This report proves that under the time step condition Delta t vertical bar Lambda vertical bar < 1 (vertical bar.vertical bar = Euclidean norm) suggested by root condition analysis and necessary for stability, all modes of the CrankNicolson Leap-Frog (CNLF) approximate solution to the system du/dt + Au + Lambda u = 0, fort > 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.