Theta method ($y_{n+1}-y_n=h[(1-a)f(y_n)+af(y_{n+1})]$)

This one-step family interpolates between the Forward Euler method at $a=0$, the trapezoidal rule at $a=1/2$, and the Backward Euler method at $a=1$. Its stability function is $R(z)=[1+(1-a)z]/[1-az]$. It is first order except at one half, where it is second order. Applying it to spatial diffusion gives the 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.

At $a=1/2$, arbitrarily stiff heat modes approach amplification $-1$: the Crank-Nicolson method is contractive but does not rapidly damp unresolved rough components. The Backward Euler method instead has limit zero. Thus unconditional linear stability and strong stiff decay are different requirements. High smooth-data consistency order alone does not imply a uniform high-order estimate down to the initial instant for arbitrary L2 initial conditions.