TY - JOUR
T1 -  Instability of Crank-Nicolson Leap-Frog for Nonautonomous Systems
AU - WILLIAM LAYTON, AZIZ TAKHIROV AND MYRON SUSSMAN
JO - International Journal of Numerical Analysis Modeling Series B
VL - 3
SP - 289
EP - 298
PY - 2014
DA - 2014/05
SN - 5
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnamb/235.html
KW - partitioned methods
KW - energy stability
AB - The implicit-explicit combination of Crank-Nicolson and Leap-Frog methods is widely used for atmosphere, ocean and climate simulations. Its stability under a CFL condition in the
autonomous case was proven by Fourier methods in 1962 and by energy methods for autonomous
systems in 2012. We provide an energy estimate showing that solution energy can grow with time
in the nonautonomous case, with worst case rate proportional to time step size. We present two
constructions showing that this worst case growth rate is attained for a sequence of timesteps
&#916t &#8594 0. The construction exhibiting this growth for leapfrog is for a problem with a periodic
coefficient.