= Theta diffusion stability criterion
{title2=$G(s)=[1-(1-a)s]/[1+as],\quad s\geq0$}
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 $a\geq1/2$. For $a<1/2$, contraction holds up to $s\leq2/(1-2a)$, with a positive denominator in that range. Orthogonality is what transfers these modal bounds to a full <L2 norm> estimate.
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