4.3: Example - the simple harmonic oscillator
- Page ID
- 219160
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Our friend the simple harmonic oscillator is also easy to implement using Heun’s method. The result of running Heun’s method using a larger step size than used for both forward and backward Euler is shown in 4.4. Compared to both forward and backward Euler, the oscillation neither grows nor decays significantly over the 5 second simulation time – at least on the scale of 4.4. This suggests Heun’s method is a superior algorithm to both forward and backward Euler. The better accuracy is a result of using slope information from both time \(t_n\) as well as \(t_{n+1}\) – this gives the method an \(h^2\) order. Nonetheless, all is not rosy – see 4.5 which plots the difference between the computed and the analytic solution. The difference evidently grows exponentially, although the magnitude is not as bad as with the Euler methods.
