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
KW - Crank-Nicolson Leap-Frog
KW - CNLF
KW - unstable mode
KW - computational mode
AB - This report proves that under the time step condition Δt|Λ| ‹ 1 (|⋅| = 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 + Λu = 0; for t > 0 and u(0) = u_0;
where A + A^T is symmetric positive definite and Λ is skew symmetric, are asymptotically stable.
This result gives a sufficient stability condition for non-commutative A and Λ, 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.