@Article{IJNAM-13-753, author = {}, title = {The Unstable Mode in the Crank-Nicolson Leap-Frog Method Is Stable}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {5}, pages = {753--762}, 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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/463.html} }