= Solution
A linear <multistep method> for $y'=f(t,y)$ has the form
$$
\sum_{j=0}^s\alpha_jy_{n+j}
=h\sum_{j=0}^s\beta_jf_{n+j},
\qquad \alpha_s=1.
$$
Its <characteristic polynomials of a linear multistep method> are $\rho(\zeta)=\sum_j\alpha_j\zeta^j$ and $\sigma(\zeta)=\sum_j\beta_j\zeta^j$. Substitution of the exact solution, or equivalently expansion of $\rho(e^z)-z\sigma(e^z)$ at $z=0$, gives the <order conditions for a linear multistep method>. The method has order $p$ when
$$
\rho(e^z)-z\sigma(e^z)=O(z^{p+1}),
$$
with a nonzero coefficient at the next power. Its one-step defect is then $O(h^{p+1})$, while its global error is $O(h^p)$ when stability prevents accumulation from being amplified.
Order alone does not ensure convergence because the recurrence may contain growing parasitic modes. <Zero-stability> is the <root condition for a multistep method>: every root of $\rho$ lies in $|\zeta|\leq1$, and every root on the unit circle is simple. Consistency is the first-order condition
$$
\rho(1)=0,
\qquad
\rho'(1)=\sigma(1).
$$
The <Dahlquist equivalence theorem> states that a consistent linear multistep method is convergent if and only if it is zero-stable. For example, the leapfrog method $y_{n+2}-y_n=2hf_{n+1}$ has $\rho(\zeta)=\zeta^2-1$; its roots $\pm1$ are simple, so this second-order method is convergent, although its parasitic $-1$ mode can oscillate.
To study stiff decay, apply the method to $y'=\lambda y$ and put $z=h\lambda$. The amplification factors are the roots of
$$
\rho(\zeta)-z\sigma(\zeta)=0.
$$
The <linear stability domain> consists of the $z$ for which these roots satisfy the strict interior condition appropriate away from the boundary, with simple unit roots where boundary stability is admitted. The method is <A-stable> when this domain contains the left half-plane, so every mode with $\operatorname{Re}\lambda<0$ remains bounded for every positive step size. The <Second Dahlquist barrier> says that an A-stable multistep method has order at most two.
The main families illustrate the tradeoffs. The explicit Adams-Bashforth methods interpolate past derivative values and are inexpensive, but their stability domains are bounded. The implicit Adams-Moulton methods include the new derivative value; the one-step second-order member is the <trapezoidal rule>, which is A-stable. The backward differentiation formulas interpolate past solution values and differentiate the interpolant; the first two are A-stable, while higher orders retain useful sectors of the left half-plane. Implicit methods require a nonlinear solve at each step, commonly initialized by an explicit predictor to form a predictor-corrector pair. Thus order controls consistency error, the root condition controls convergence as $h\to0$, and the amplification polynomial controls whether a convergent method remains useful on stiff equations.
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