A pendulum in a teaching laboratory is modelled at small amplitude as with

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Simulating classical dynamical systems

Computational Physics, Ch Stepping the equations of motion: four integrators measured

Problem

A pendulum in a teaching laboratory is modelled at small amplitude as θ¨=−ω02θ\ddot\theta = -\omega_0^2\theta with ω0=4.0 s−1\omega_0 = 4.0\,\mathrm{s^{-1}}, released from θ0=0.10 rad\theta_0 = 0.10\,\mathrm{rad} at rest. It is integrated by velocity Verlet, and the angle at t=10.0 st = 10.0\,\mathrm{s} is read off at five steps, each half the last. The values of θ(10.0 s)\theta(10.0\,\mathrm{s}), in radians, are −0.0715305-0.0715305 at Δt=0.05 s\Delta t = 0.05\,\mathrm{s}, then −0.0679277-0.0679277, −0.0670038-0.0670038, −0.0667714-0.0667714 and −0.0667132-0.0667132 at Δt=0.025\Delta t = 0.025, 0.01250.0125, 0.006250.00625 and 0.003125 s0.003125\,\mathrm{s}. ((a)) Form the successive changes and their ratios, and state the measured order of the method on this quantity. ((b)) The measurement this feeds is quoted to 1×10−5 rad1\times10^{-5}\,\mathrm{rad}, and the step is to be chosen so that one further halving would change θ(10.0 s)\theta(10.0\,\mathrm{s}) by less than that. No run finer than the last row exists. Use the measured order to find the largest step in this halving sequence that meets the requirement, and say how many steps the 10.0 s10.0\,\mathrm{s} run then takes. ((c)) The same model is then to be run for 10610^6 periods to study its energy. Say which of velocity Verlet and fourth-order Runge-Kutta you would run for that, and name the quantity you would watch over the whole run that the halving table above cannot tell you about.

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