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Figure 7-7. A closer look
As you can see from these figures, it's impossible to discern the curves for the improved
Euler and Runge-Kutta methods from the exact solution because they fall almost right
on top of each other. These results clearly show the improvement in accuracy over the
basic Euler method, whose curve is distinct from the other three. Over the interval from
6.5 to 8.5 seconds, the average truncation error is 1.72%, 0.03%, and 3.6×10 −6 % for
Euler's method, the improved Euler method, the Runge-Kutta method, respectively. It
is obvious, based on these results, that for this problem, the Runge-Kutta method yields
substantially better results for a given step size than the other two methods. Of course,
you pay for this accuracy, since you have several more computations per step in the
Runge-Kutta method.
Both of these methods are generally more stable than Euler's method, which is a huge
benefit in real-time applications. Recall our discussion earlier about the stability of Eu‐
ler's method. Figure 7-5 showed the results of applying Euler's method to an oscillating
dynamical system. There, the motion results that should be sinusoidal were wildly er‐
ratic (i.e., unstable). Applying the improved Euler method, or the Runge-Kutta method,
to the same problem yields stable results, as shown in Figure 7-8 .
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