 Solution

The characteristic polynomial is ${\lambda }^2+2\gamma \lambda +k^2=0$. Since $0<\gamma <|k|$,Forward Euler method has amplification eigenvalues $g_{\pm }=1+h{\lambda }_{\pm }$. Their squared modulus isThusStrict inequality gives numerical damping; at the upper boundary the amplitude is constant despite physical decay, so that boundary is a poor accuracy choice. In the undamped limit $\gamma =0$, $|g{|}^2=1+k^2{h}^2>1$ for every $h>0$: forward Euler method is unstable for any nonzero step on an undamped harmonic oscillator.