@Article{IJNAM-13-5,
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 = {<p style="text-align: justify;">This report proves that under the time step condition&nbsp; $\bigtriangleup t|\Lambda|&lt;1$(| $\cdot$ | = 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<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $\frac{du}{dt}+ Au + \Lambda u = 0$, for $t &gt; 0$ and $u(0) = u_0$, <br/>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.</p>},
issn = {2617-8710},
doi = {https://doi.org/2016-IJNAM-463},
url = {https://global-sci.com/article/83262/the-unstable-mode-in-the-crank-nicolson-leap-frog-method-is-stable}
}