Volume 5, Issue 3
Instability of Crank-Nicolson Leap-Frog for Nonautonomous Systems

WILLIAM LAYTON, AZIZ TAKHIROV AND MYRON SUSSMAN

Int. J. Numer. Anal. Mod. B, 5 (2014), pp. 289-298

Published online: 2014-05

Export citation
  • Abstract
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 Δt → 0. The construction exhibiting this growth for leapfrog is for a problem with a periodic coefficient.
  • AMS Subject Headings

65M12 65J08

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{IJNAMB-5-289, author = {WILLIAM LAYTON, AZIZ TAKHIROV AND MYRON SUSSMAN}, title = { Instability of Crank-Nicolson Leap-Frog for Nonautonomous Systems}, journal = {International Journal of Numerical Analysis Modeling Series B}, year = {2014}, volume = {5}, number = {3}, pages = {289--298}, abstract = {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 Δt → 0. The construction exhibiting this growth for leapfrog is for a problem with a periodic coefficient.}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnamb/235.html} }
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 Δt → 0. The construction exhibiting this growth for leapfrog is for a problem with a periodic coefficient.
WILLIAM LAYTON, AZIZ TAKHIROV AND MYRON SUSSMAN. (1970). Instability of Crank-Nicolson Leap-Frog for Nonautonomous Systems. International Journal of Numerical Analysis Modeling Series B. 5 (3). 289-298. doi:
Copy to clipboard
The citation has been copied to your clipboard