TY - JOUR
T1 - The Unstable Mode in the Crank-Nicolson Leap-Frog Method Is Stable
JO - International Journal of Numerical Analysis and Modeling
VL - 5
SP - 753
EP - 762
PY - 2016
DA - 2016/09
SN - 13
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/463.html
KW - IMEX method, Crank-Nicolson Leap-Frog, CNLF, unstable mode, computational mode.
AB - <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>