 Zero-stability

Zero-stability controls whether small perturbations are amplified uniformly as a fixed time interval is covered by more and more steps. For a constant-coefficient linear multistep method, the zero-step recurrence isZero-stability requires a bound ${\max }_{0\le n\le T/h}|e_n|\le C{\max }_{0\le j<k}|e_j|$, with $C$ independent of small $h$ and the starting perturbation. The root condition for a multistep method characterizes this uniform boundedness.^[ref]

For $\rho (\zeta )={\zeta }^2-1$, every solution is $e_n=A+B(-1{)}^n$, so perturbations remain bounded even though the alternating component does not decay. For $\rho (\zeta )=(\zeta -1{)}^2$, the mode $e_n=n$ grows like $1/h$ at fixed positive time and violates zero-stability. This is distinct from absolute stability, which tests nonzero $z=h\lambda$ and can constrain a particular step size.