full
search.png

Search

close.png

Cancel

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

N. Hurl, W. Layton, Y. Li & M. Moraiti

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

Published online: 2016-09

Preview Purchase PDF 1903 25136

Cited by

google scholar semantic scholar
Export citation
  • 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.

  • AMS Subject Headings

65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{IJNAM-13-753, author = {N. Hurl, W. Layton, Y. Li and M. Moraiti}, 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 AU - N. Hurl, W. Layton, Y. Li & M. Moraiti 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 and M. Moraiti. (2016). The Unstable Mode in the Crank-Nicolson Leap-Frog Method Is Stable. International Journal of Numerical Analysis and Modeling. 13 (5). 753-762. doi:
Copy to clipboard
BibteX RIS TXT
The citation has been copied to your clipboard