5.6

Let \( \lambda = i\omega \)

For RK4:
\[\sigma = 1+\lambda h+\frac{1}{2}\lambda^2h^2+\frac{1}{6}\lambda^3h^3+ \frac{1}{24}\lambda^4h^4 \]
\[\sigma = 1+i\omega h-\frac{1}{2}(\omega^2h^2)-i\frac{1}{6}(\omega^3h^3)+ \frac{1}{24}(\omega^4h^h) \]
\[\sigma_r = 1-\frac{1}{2}(\omega^2h^2)+\frac{1}{24}(\omega^4h^4) \quad\quad \sigma_i = i\omega h-i\frac{1}{6}(\omega^3h^3) \]
The global error is the difference between the ODE and O\( \Delta \)E solution at time \( T=Nh \)
\[ Er_\lambda = e^{\lambda Nh} -\big(\sigma(\lambda h)\big)^N \]
the amplitude error
\[Er_a = 1-\left(\sqrt{\sigma_r^2+\sigma_i^2}\right)^N \]
\[\sigma_r^2 = 1-(\omega h)^2+\frac{1}{3}(\omega h)^4-\frac{1}{24}(\omega h)^6 +\frac{1}{576}(\omega h)^8 \]
\[ \sigma_i^2 = (\omega h)^2 -\frac{1}{3}(\omega h)^4+\frac{1}{36}(\omega h)^6 \]
\[Er_a = 1-\left(\frac{1}{24}\sqrt{(\omega h)^8 - 8(\omega h)^6 + 576}\right)^N \]
and the phase error (in degrees)
\[Er_\omega = N\frac{180}{\pi}\Bigg(\omega h-\tan^{-1}\left(\frac{\sigma_i}{\sigma_r}\right)\Bigg) \]
\[Er_\omega = N\frac{180}{\pi}\Bigg(\omega h-\tan^{-1}\left(\frac{24\omega h -4(\omega h)^3}{(\omega h)^4-12(\omega h)^2 + 24}\right)\Bigg) \]
The errors at \( N = 300 \) and \( \omega h = 0.10 \) are
\begin{align*}
\text{Amplitude: }& 1 + Er_a = 1.000002 \\
\text{Phase: }& 0.001427 deg
\end{align*}

For RK3:
\[\sigma = 1+\lambda h+\frac{1}{2}\lambda^2h^2+\frac{1}{6}\lambda^3h^3 \]
\[ \sigma_r = 1-\frac{1}{2}(\omega h)^2 \quad\quad \sigma_i=\omega h-\frac{1}{6}(\omega h^6) \]
\[ Er_a = 1-\Big(\frac{1}{6}\sqrt{(\omega h)^6-3(\omega h)^4+36} \Big)^N\]
\[Er_\omega=N\frac{180}{\pi}\Bigg(\omega h-\tan^{-1}\left(\frac{6\omega h-(\omega h)^3}{6-3(\omega h)^2}\right) \Bigg) \]
The errors at \( N=400 \) and \( \omega h=0.075 \) are
\begin{align*}
\text{Amplitude: }& 1 + Er_a = 1.000526 \\
\text{Phase: }& -0.0018124 deg
\end{align*}

For RK2:
\[\sigma = 1+\lambda h+\frac{1}{2}\lambda^2h^2 \]
\[ \sigma_r = 1-\frac{1}{2}(\omega h)^2 \quad\quad \sigma_i=\omega h \]
\[ Er_a = 1-\Big(\frac{1}{2}\sqrt{(4+(\omega h)^4} \Big)^N\]
\[Er_\omega=N\frac{180}{\pi}\Bigg(\omega h-\tan^{-1}\left(\frac{2\omega h}{2-(\omega h)^2}\right) \Bigg) \]
The errors at \( N=600 \) and \( \omega h=0.05 \) are
\begin{align*}
\text{Amplitude: }& 1 + Er_a = 0.999531 \\
\text{Phase: }& -0.71566 deg
\end{align*}

I'm getting 0.00143 deg for RK4 as well.