= Solution
Consider an <initial value problem> $y'=f(t,y)$ on $[0,T]$, with a <Lipschitz continuous> <vector field> in the solution variable and enough smoothness for the stated error expansions. <Convergence of a numerical method> means
$$
\max_{t_n\leq T}\|Y_n-y(t_n)\|\longrightarrow0\qquad(h\to0),
$$
with consistent starting values. An <order of a numerical method> $p$ quantifies this approximation: with sufficiently accurate starting data and the required stability, the <global error> is $O(h^p)$. <Stability of a numerical method> controls the amplification and accumulation of perturbations; it is the link between a small local defect and small global error.
For a one-step formula $Y_{n+1}=\Phi_h(t_n,Y_n)$, define the unscaled <local truncation error>
$$
d_{n+1}=y(t_{n+1})-\Phi_h(t_n,y(t_n)).
$$
The method has local order $p$ if $\|d_{n+1}\|\leq Ch^{p+1}$ uniformly along sufficiently smooth solutions. If the defect is divided by $h$, its order is instead $O(h^p)$; the convention must be specified. Assume also the perturbation estimate
$$
\|\Phi_h(t,u)-\Phi_h(t,v)\|\leq(1+L_\Phi h)\|u-v\|.
$$
The <global error> $e_n=Y_n-y(t_n)$ then satisfies
$$
\|e_{n+1}\|\leq(1+L_\Phi h)\|e_n\|+Ch^{p+1}.
$$
Iterating, or applying the <discrete Gronwall inequality>, gives
$$
\boxed{\max_{nh\leq T}\|e_n\|\leq e^{L_\Phi T}\bigl(\|e_0\|+CT h^p\bigr).}
$$
There are $O(1/h)$ steps on a fixed interval, so stable propagation turns $O(h^{p+1})$ local defects into $O(h^p)$ accumulated error. Starting error $O(h^p)$ is needed to retain order $p$. This explains why matching an exact <Taylor expansion> is insufficient unless errors are controlled during repeated steps.
For a fixed-coefficient <linear multistep method>, write
$$
\sum_{j=0}^s\alpha_jY_{n+j}=h\sum_{j=0}^s\beta_jf(t_{n+j},Y_{n+j}),\quad
\rho(\zeta)=\sum_j\alpha_j\zeta^j,\quad
\sigma(\zeta)=\sum_j\beta_j\zeta^j.
$$
The <consistency of a numerical method> conditions are $\rho(1)=0$ and $\rho'(1)=\sigma(1)$. The higher <order conditions for a linear multistep method> are
$$
\sum_j\alpha_jj^q=q\sum_j\beta_jj^{q-1},\qquad q=0,\ldots,p,
$$
with right side zero for $q=0$; exact order $p$ means first failure at $q=p+1$. They come from inserting the exact solution and comparing its <Taylor expansion>.
Errors at $h=0$ follow the homogeneous recurrence determined by $\rho$. <Zero-stability> is equivalent to the <root condition for a multistep method>: all <roots of a polynomial> have <modulus> at most one, and every unit-modulus root is simple. Roots outside the disk give exponential growth in the number of steps; repeated unit roots give polynomial growth in that number. Repeated interior roots are harmless because their polynomial factors are dominated by decay. Under the usual <Lipschitz continuity> and initialization assumptions, the <Dahlquist equivalence theorem> gives
$$
\boxed{\text{convergence}=\text{consistency plus zero-stability}.}
$$
For a zero-stable order-$p$ <linear multistep method>, local defects and starting errors of the stated orders yield <global error> $O(h^p)$ by a recurrence bound and the <discrete Gronwall inequality>. Every required starting value matters.
A concrete failure of “consistency implies convergence” is
$$
Y_{n+2}-3Y_{n+1}+2Y_n=-h f(t_n,Y_n).
$$
Here $\rho=(\zeta-1)(\zeta-2)$ and $\sigma=-1$, so $\rho(1)=0$ and $\rho'(1)=\sigma(1)=-1$: the method is consistent, of order one. But its root $2$ violates <zero-stability>. For $y'=0$ with exact solution zero and starting values $Y_0=0$, $Y_1=h^2$, the computed solution is $Y_n=h^2(2^n-1)$. These starting errors vanish, yet at $n\approx T/h$ the error diverges. Local accuracy cannot compensate for an unstable recurrence.
A different meaning of <stability> concerns a fixed step on decaying equations. The <Dahlquist test equation> $y'=\lambda y$ introduces $z=h\lambda$. A one-step method gives $Y_{n+1}=R(z)Y_n$ and is absolutely stable when $|R(z)|\leq1$. For a <linear multistep method>, the corresponding <amplification polynomial of a multistep method> is $\rho(\zeta)-z\sigma(\zeta)$, and every root must satisfy the bounded root condition, not just the root approximating $e^z$. Strict asymptotic decay requires the roots to have <modulus> less than one for $\operatorname{Re}z<0$. This distinction matters at reducible methods with an undamped parasitic mode, as in question 1(c).
The <Forward Euler method> has order one and $R(z)=1+z$, so its <linear stability domain> is $|1+z|\leq1$. For $y'=-\kappa y$, with $\kappa>0$, it requires $0\leq h\kappa\leq2$. It converges as $h\to0$, yet an excessively large step can produce growing oscillations on an exactly decaying problem. The <Backward Euler method> has the same order, but $R(z)=(1-z)^{-1}$ makes it <A-stable> and <L-stable>. For a <stiff differential equation>, fast decay can impose a severe explicit time-step restriction even when accuracy of the slow dynamics would permit a much larger step.
The <trapezoidal rule> has order two and is <A-stable>, but is not <L-stable>, since its <stability function> tends to $-1$ along the negative real axis. The <Second Dahlquist barrier> limits an irreducible <A-stable> <linear multistep method> to order at most two. This does not limit implicit <Runge-Kutta methods>: question 4's <Lobatto IIIA method> is fourth-order and <A-stable>. Conversely, question 1's fourth-order member is convergent but not <A-stable>. These examples show that <order of a numerical method>, <zero-stability> and <A-stability> answer different questions.
For nonlinear dissipative problems, <B-stability> controls distances between numerical solutions; <algebraic stability of a Runge-Kutta method> is a sufficient coefficient criterion. Question 4 shows that <A-stability> alone does not imply that coefficient criterion. Variable time steps require additional analysis of step ratios and error propagation, and stiff problems may require uniform-in-stiffness error estimates beyond the fixed-problem convergence result. \b[The useful chain is local accuracy plus the appropriate perturbation bound, giving global accuracy; absolute and nonlinear stability then determine which practical time steps preserve the intended dynamics.]
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