 Theta diffusion stability criterion

On an orthogonal diffusion eigenmode, the theta method has the displayed amplification with $s=-\Delta t\lambda$. The identity $(1+as{)}^2-[1-(1-a)s{]}^2=2s+(2a-1)s^2$ proves contraction at all nonnegative $s$ exactly when $a\ge 1/2$. For $a<1/2$, contraction holds up to $s\le 2/(1-2a)$, with a positive denominator in that range. Orthogonality is what transfers these modal bounds to a full L2 norm estimate.