Strict linear stability domain (${\mathcal{D}}_{\mathrm{decay}}=\{z:y_n\to 0\}$)

The strict decay convention for a linear stability domain demands that every numerical mode tends to zero on the Dahlquist test equation. For a one-step stability function it requires $|R(z)|<1$; for a multistep method it requires every amplification root to have modulus below one. This differs from a boundedness convention that allows simple unit-circle roots, and from mesh-uniform finite-time numerical stability. Cambridge numerical analysis notes use the decay convention. Explicitly specifying the convention is essential for methods with undamped modes.