Volume 13, Issue 5
The Unstable Mode in the Crank-Nicolson Leap-Frog Method Is Stable

Int. J. Numer. Anal. Mod., 13 (2016), pp. 753-762.

Published online: 2016-09

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• 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.

• Keywords

IMEX method, Crank-Nicolson Leap-Frog, CNLF, unstable mode, computational mode.

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@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} }
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 -

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.

N. Hurl, W. Layton, Y. Li & M. Moraiti. (1970). The Unstable Mode in the Crank-Nicolson Leap-Frog Method Is Stable. International Journal of Numerical Analysis and Modeling. 13 (5). 753-762. doi:
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