= Rough-data damping distinction for the theta method
{title2=$G(s)\to-(1-a)/a\quad(s\to\infty,\ a>0)$}
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>.
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