= Solution
A <stiff differential equation> can contain rapidly decaying modes as well as a slowly changing solution of interest. An explicit scheme may need a very small step solely to keep those already small fast modes from numerical growth. <A-stability> addresses this restriction through the <Dahlquist test equation> $y'=\lambda y$, whose exact solution decays when $\operatorname{Re}\lambda<0$.
For a one-step method, write $y_{n+1}=R(z)y_n$, $z=h\lambda$. Its <linear stability domain> consists of the points where the update is defined and $|R(z)|\leq1$. \b[<A-stability> means that this domain contains the closed left half-plane.] This is a scalar linear stability property, not an order condition, an existence theorem for an implicit solve, or a general nonlinear contractivity assertion.
For a rational approximation to the exponential, consistency requires $R(z)=1+z+O(z^2)$, and order $p$ requires $R(z)-e^z=O(z^{p+1})$. Poles must be absent from the relevant half-plane. The <Forward Euler method> has $R=1+z$ and the disk $|1+z|\leq1$, so negative real modes require $h|\lambda|\leq2$; it is not <A-stable>. The <Backward Euler method> has $R=(1-z)^{-1}$, and $|1-z|\geq1$ on the left half-plane, so it is <A-stable>. The <trapezoidal rule> and the <implicit midpoint rule> both have, on the scalar linear problem,
$$
R(z)=\frac{1+z/2}{1-z/2},\qquad
|1-z/2|^2-|1+z/2|^2=-2\operatorname{Re}z.
$$
They are second-order and <A-stable> despite being different nonlinear methods. No nonconstant polynomial approximation is <A-stable>, because its modulus grows without bound on the negative real axis. In particular every nontrivial explicit <Runge-Kutta method> has a bounded-step stability restriction.
<A-stability> prevents growth but need not eliminate extremely stiff modes: the trapezoidal factor tends to minus one as $z\to-\infty$. <L-stability> additionally requires $R(z)\to0$ within the left half-plane. Backward Euler is <L-stable>; trapezoidal and implicit midpoint are not. The distinction explains persistent alternating transients in otherwise stable Crank-Nicolson calculations. Accuracy still constrains useful step sizes even for an <A-stable> method.
A <linear multistep method> is characterized by polynomials $\rho,\sigma$, and its scalar modes satisfy $\rho(\zeta)-z\sigma(\zeta)=0$. There is generally no single scalar amplification factor: every root must lie in the closed unit disk and unit roots must be simple. At $z=0$, this is the <zero-stability> root condition. Consistency plus <zero-stability> gives convergence by the <Dahlquist equivalence theorem>, with suitable starting values. <A-stability> demands the amplification root condition throughout the left half-plane, including the zero-step condition. The <Second Dahlquist barrier> says that an irreducible <A-stable> <linear multistep method> has order at most two, and an explicit <linear multistep method> cannot be <A-stable>.
Examples include backward Euler, trapezoidal, and the second-order <backward differentiation formula>
$$
3y_{n+2}-4y_{n+1}+y_n=2hf(y_{n+2}).
$$
Its zero-step roots are $1,1/3$. Its unit-circle boundary quotient has real part $(1-\cos\theta)^2\geq0$, and the leading coefficient is nonzero on the left half-plane, giving <A-stability> by the same continuation argument as in Question 2. Conversely the third-order member of that question fails <A-stability>. A formal high order without <zero-stability>, or after silently cancelling a problematic unit factor, does not evade the barrier.
For an <implicit Runge-Kutta method>, the stages on the test equation satisfy $Y=e\,y_n+zAY$. Whenever these stages are solvable, its <stability function> is
$$
\boxed{R(z)=1+z\,b^T(I-zA)^{-1}e.}
$$
This rational function is used to test scalar <A-stability>; the <Butcher order conditions> separately establish nonlinear order. The <Lobatto IIIA method> in Question 3 is an <A-stable> fourth-order example with $R$ the diagonal degree-two Padé approximant. More generally the $s$-stage <Gauss--Legendre Runge-Kutta method> has order $2s$ and is <A-stable>, with the diagonal Padé approximation to $e^z$; its stiff-limit factor is nonzero. The $s$-stage <Radau IIA method> has order $2s-1$ and is <L-stable>. Thus Runge-Kutta stages permit arbitrarily high <A-stable> order, unlike irreducible linear multistep formulas.
For a <normal matrix>, scalar amplification bounds transfer through unitary diagonalization. For a <non-normal matrix>, <eigenvalue> information alone does not assert Euclidean contraction: transient growth and eigenvector conditioning can matter. Direct <energy method> arguments, such as the dissipative Cayley-transform proof in Question 1, use the symmetric part and are more informative in that setting.
For nonlinear dissipative vector fields, the stronger <B-stability> property asks that distances between two numerical solutions not increase. A standard sufficient condition for <Runge-Kutta methods> is <algebraic stability>: $b_i\geq0$ and
$$
(b_i a_{ij}+b_j a_{ji}-b_i b_j)_{ij}
$$
is positive semidefinite. It follows from the <Runge-Kutta contractivity identity> when the implicit stages exist. The three-stage Lobatto IIIA method is <A-stable> but has the first diagonal entry of that <matrix> equal to $-1/36$, so scalar <A-stability> should not be confused with that sufficient nonlinear condition. <Stage solvability of an implicit Runge-Kutta method> remains a separate issue and computational cost of the implicit equations must be considered. \b[<A-stability> removes a scalar decay-mode stability restriction; order, stiff damping, nonlinear stability and solvability remain distinct properties.]
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