A lightly damped oscillator obeys with and

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Ordinary differential equations: order, stability and stiffness

Problem

A lightly damped oscillator obeys

\ddot{y} + 2\zeta\omega\dot{y} + \omega^2 y = 0

with

\omega = 5\,\mathrm{rad\,s^{-1}}

and

\zeta = 0.1

. Written as a first-order system, its Jacobian has eigenvalues

\lambda = -\zeta\omega \pm i\omega\sqrt{1-\zeta^2}

. Classical RK4's stability region reaches out to

h|\lambda| = 2.83

near the imaginary axis. Using this limit as the relevant boundary for these nearly-oscillatory eigenvalues, estimate the largest stable step

h

for RK4 applied to this oscillator.

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