Instability of Crank-Nicolson Leap-Frog for Nonautonomous Systems

stability in the autonomous, scalar case was proven in 1963 in Johansson and Kreiss (9), see also (4), and for non-commuting, autonomous systems in 2012 (12), see also (17) for background. We prove herein weak instability in the nonautonomous case. The extension of stability for ODEs from autonomous to nonautonomous (with test problem y′ = λ (t)y) has a rich history. Dahlquist (3) proved that an A-stable method is similarly stable for y′ = λ (t)y when Re(λ (t)) 6 0, further developed in (13). For the corresponding AN-stability theory for RungeKutta methods, see Hundsdorfer and Stetter (7). For non-A-stable multi-step methods, nonautonomous stability theory was recently developed in Boutelje and Hill (2). Their theory gives conditions under which a method will be stable for y′ = λ (t)y and under which it will be unstable. For example, given a linear multistep method for y′ = λ (t)y , let ρ(z),σ(z) be the complex polynomials associated with the method in a standard way and form