Rough-data damping distinction for the theta method ($G(s)\to -(1-a)/a(s\to \infty ,a>0)$)

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.