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

Zichen Deng

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

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  • 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.

  • History

Published online: 2014-05

  • AMS Subject Headings

65M12, 65J08

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