Solution
= Solution
A fixed-step <linear multistep method> for $y'=f(t,y)$ uses several previous solution and derivative values:
$$
\sum_{j=0}^s\alpha_jy_{n+j}
=k\sum_{j=0}^s\beta_jf(t_{n+j},y_{n+j}),\qquad \alpha_s\ne0.
$$
The coefficients are fixed independently of the step size. It is explicit when $\beta_s=0$ and implicit otherwise. Its <characteristic polynomials of a linear multistep method> are $\rho(w)=\sum_j\alpha_jw^j$ and $\sigma(w)=\sum_j\beta_jw^j$. The <contraction mapping theorem> gives a uniquely solvable implicit step under a <Lipschitz continuity> bound $L$ whenever $k|\beta_s|L<|\alpha_s|$, by contraction of the new-value equation on the appropriate solution region. A suitable starting procedure must supply the first $s$ values; the method alone does not determine them.
Order is defined by inserting a smooth exact solution into the recurrence. If the residual is $O(k^{p+1})$ and its first nonzero coefficient occurs at that degree, the method has order $p$; dividing the residual by $k$ gives an $O(k^p)$ <local truncation error>. <Taylor expansion> gives the exact <order conditions for a linear multistep method>
$$
\sum_j\alpha_jj^q=q\sum_j\beta_jj^{q-1}\quad(0\leq q\leq p),
$$
with the right side interpreted as zero at $q=0$, and failure at $q=p+1$ for exact order. Equivalently,
$$
\rho(e^z)-z\sigma(e^z)=O(z^{p+1}).
$$
The basic <consistency of a numerical method> conditions are $\rho(1)=0$, $\rho'(1)=\sigma(1)$; for the ordinary nondegenerate consistent formulation this common value is nonzero. Exactness on constants and linear functions does not control propagation of numerical perturbations.
That propagation is the role of <zero-stability>. On $y'=0$ the homogeneous recurrence is $\rho(E)y_n=0$. Its modes are $n^r\xi^n$, with powers of $n$ determined by root multiplicities. The theorem on the <root condition for a multistep method> states that <zero-stability> is equivalent to every root of $\rho$ having modulus at most one, with every unit-modulus root simple. Interior repeated roots are permitted, since geometric decay dominates their <polynomial> factors. An exterior root amplifies perturbations geometrically; a repeated unit root amplifies them polynomially and violates a step-independent perturbation bound.
The <Dahlquist equivalence theorem> states that a fixed-step consistent <linear multistep method> is convergent exactly when it is <zero-stable>. This concerns <ordinary differential equations> with a uniformly <Lipschitz continuous> right-hand side on the solution region over a fixed finite interval, convergent starting values, and the well-defined nearby branch of each small-step implicit solve. For order $p$ <convergence of a numerical method>, the solution must have the required smoothness and the starting errors must be $O(k^p)$.
The mechanism can be sketched explicitly. The <root condition for a multistep method> bounds the recurrence's discrete <Green function>. For an order-$p$ exact-solution defect $d_n=O(k^{p+1})$, the <Lipschitz continuity> error equation then gives an estimate of the form
$$
E_n\leq C\left(E_{\mathrm{start}}+\sum_{j<n}\|d_j\|\right)
+Ck\sum_{j\leq n}E_j.
$$
The current-step term can be absorbed for small $k$. The <discrete Gronwall inequality> says that a bound $E_n\leq A+Bk\sum_{j<n}E_j$ implies $E_n\leq A(1+Bk)^n\leq Ae^{Bt_n}$. There are $O(k^{-1})$ defects on a fixed interval, so their sum is $O(k^p)$ and this yields the claimed global order. Conversely, testing $y'=0$ with vanishing starting perturbations in exterior-root or repeated-unit-root modes proves that <convergence of a numerical method> fails without the <root condition for a multistep method>.
For example, the two-step <Adams-Bashforth method> has
$$
y_{n+2}-y_{n+1}=k(3f_{n+1}-f_n)/2,
\qquad \rho=w(w-1),\quad\sigma=(3w-1)/2.
$$
It is explicit, order two and <zero-stable>, hence convergent with order-two starting values. By contrast, a recurrence with $\rho=(w-1)^2$ has solutions $y_n=y_0+n(y_1-y_0)$ on the zero differential equation. Starting differences $\sqrt k$ tend to zero yet become unbounded at a fixed positive time, since $n$ is of order $1/k$. A small local defect or formal high order cannot rescue this mode. Canceling a common factor may remove it only by changing the allowed starting relations, as the parameter endpoint in Question 2 illustrates.
<Absolute stability> addresses a different limit: apply the method to the <Dahlquist test equation> $y'=\lambda y$ and put $z=k\lambda$. The amplification roots satisfy
$$
P_z(w)=\rho(w)-z\sigma(w)=0.
$$
All roots must satisfy the disk condition, with unit roots simple; one cannot inspect only the root that tends to one as $z\to0$. The degree and solvability of the recurrence must also be preserved. The resulting set of $z$ is the domain of <absolute stability>. <A-stability> means that it includes the whole closed left half-plane, so the numerical solutions of every <Dahlquist test equation> with $\operatorname{Re}\lambda\leq0$ are stable for every positive step size. It is a linear test-equation property, not a proof of arbitrary nonlinear contractivity or high accuracy at large steps.
<Forward Euler method> has order one and factor $1+z$, so its stability region is the disk $|1+z|\leq1$. <Backward Euler method> has factor $(1-z)^{-1}$ and is <A-stable>. The <trapezoidal rule> is order two with factor $(1+z/2)/(1-z/2)$; its modulus is at most one throughout the left half-plane. These one-step methods are included as the simplest multistep examples. For the two-step <Adams-Bashforth method>, the negative-real stability interval is $-1\leq z\leq0$: its <polynomial> is $w^2-(1+3z/2)w+z/2$, and the root reaches $-1$ at $z=-1$. It therefore imposes a step restriction when applied to strongly decaying modes.
The genuinely two-step <BDF2 method> is
$$
3y_{n+2}-4y_{n+1}+y_n=2k f_{n+2}.
$$
It has order two, roots $1,1/3$ at zero step and is <A-stable>. For its normalized <polynomials> $\rho=3w^2/2-2w+1/2$, $\sigma=w^2$, the unit-circle boundary locus has
$$
\operatorname{Re}\frac{\rho(e^{i\theta})}{\sigma(e^{i\theta})}
=(1-\cos\theta)^2\geq0.
$$
The leading coefficient has no zero in the left half-plane, and the roots at a small negative $z$ are both inside the disk. Root continuity then proves open-half-plane stability; on the imaginary boundary the only possible unit root is the simple root one at $z=0$. This gives a direct proof of the stated <A-stability> rather than relying on a boundary plot.
Two sharp order barriers explain the tradeoff. The <first Dahlquist barrier> states that a consistent <zero-stable> real linear $s$-step method using only first derivatives has order at most $s+1$ for odd $s$ and $s+2$ for even $s$; an explicit method has order at most $s$. The <Second Dahlquist barrier> states that an irreducible <A-stable> real <linear multistep method> has order at most two. These statements concern fixed constant coefficients and the ordinary first-derivative multistep class, not multiderivative methods or arbitrary variable-step formulations. Thus higher order alone is not a route to unconditional stiff stability. Question 2 provides a concrete illustration: its third-order member is <zero-stable> but not <A-stable>.
No nontrivial consistent explicit <multistep method> is <A-stable>. Its <amplification polynomial of a multistep method> has a fixed highest coefficient, while at least one lower coefficient grows without bound as real $z\to-\infty$. Were all roots uniformly in the <unit disk>, their <elementary symmetric polynomials> would keep all the monic coefficients bounded, a contradiction. This is the coefficient argument behind <explicit multistep methods cannot be A-stable>.
In practice, selecting a method therefore involves accuracy, startup, the presence of a <stiff differential equation>, implicit-solve cost and step-size control. <Zero-stability> handles propagation as the step tends to zero; <consistency of a numerical method> supplies a vanishing defect; <A-stability> handles the numerical solutions of all decaying <Dahlquist test equations> at finite steps. None substitutes for the others, and variable steps or PDE discretizations require additional uniform stability arguments appropriate to their setting.