= Solution
Backward <Euler method> has update $\mathbf y^{n+1}=(I-hM)^{-1}\mathbf y^n$ and amplification eigenvalues $g_\pm=(1-h\lambda_\pm)^{-1}$. Hence
$$
\boxed{|g_\pm|^2=\frac1{(1+\gamma h)^2+(k^2-\gamma^2)h^2}=\frac1{1+2\gamma h+k^2h^2}\le1}
$$
for every $h\ge0$. It is therefore unconditionally stable for this <linear system>. Even with $\gamma=0$, it damps every nonzero numerical mode: stability is obtained at the cost of artificial damping and phase error. Large $h$ does not accurately resolve the oscillation period.
Back to article page
