Topic 2: The pendulum

PHY321F — cp 2005 28
2.4 Higer-order methods
Better accuracy is achieved in a single time-step by using a higher-order
integration method. These typically use more derivative information to gain
accuracy.
A simple example is (velocity) Verlet method. (There are other Verlet methods
equivalent mathematically although not necessarily numerically).
θ i+1 = θ i + ω i ∆t + 1 2 a i (∆t) 2
ω i+1 = ω i + 1 2 (a i + a i+1 ) ∆t
This method also has the advantage of preserving energy conservation.
Aother second order method is the Euler-Richardson method which uses the
slope in the middle of the time-step for extrapolation to the next point. This
results in O(∆t) 2 accuracy, at the expense of a second force evaluation.
a i = α(θ i , ω i , t i )
ω mid = ω i + a i
∆t
2
θ mid = θ i + ω i
∆t
2
a mid = α(θ mid , ω mid , t i + ∆t
2 )
ω i+1 = ω i + a mid ∆t
θ i+1 = θ i + ω mid ∆t
This is essentially a second order Runge-Kutta method and can be derived
in several ways. It can be combined with the standard Euler step to obtain
information on the truncation error. With this the method can easily be
extended to offer adaptive control: the step-size can be varied to maintain
an error limit.
2.5 Exercises
1. Compare the Euler and Euler-Cromer methods of integration.
2. Determine the dependence of the frequency of the pendulum on its
amplitude. One possible means of analysis is to determine the power
spectrum of the oscillations from the Fourier transform of the solution.