Scalar absolute stability, A-stability, L-stability, and multistep root regions

For the scalar test equation $y^{\mathrm{\prime }}=\lambda y$, a step size $h>0$ gives $z=h\lambda$. A one-step scalar update has stability function $R$ when one step sends $y_n$ to $y_{n+1}=R(z)y_n$. If $R$ is rational, write it in reduced form; it is defined at $z$ exactly when its denominator is nonzero there. Its absolute-stability region is

${\mathcal{S}}_R=\{z\in \mathbb{C}:R(z)\text{ is defined and }\mathrm{\mid }R(z)\mathrm{\mid }\le 1\}\mathrm{.}$

The update is A-stable when the closed left half-plane $\{z:\mathrm{Re}z\le 0\}$ is contained in ${\mathcal{S}}_R$. A rational one-step update is L-stable when it is A-stable and

${\lim }_{\begin{array}{c}\mathrm{\mid }z\mathrm{\mid }\to \mathrm{\infty }\\ \mathrm{Re}z\le 0\end{array}}R(z)=0,$

where the limit means that the values tend to zero along every sequence in the closed left half-plane whose modulus tends to infinity.

For a linear multistep method with the coefficient and polynomial convention of Linear multistep methods, consistency, order, and the root condition, substitution of the scalar test equation gives the characteristic polynomial

${\pi }_z(\zeta )=\rho (\zeta )-z\sigma (\zeta )\mathrm{.}$

The multistep absolute-stability region consists of those $z$ for which the leading coefficient ${\alpha }_k-z{\beta }_k$ is nonzero and every root of ${\pi }_z$ satisfies the root condition: all roots have modulus at most one, and every root of modulus one is simple. This criterion concerns constant-step scalar test recurrences.

Absolute stability is a scalar test-equation property. It does not itself assert an accuracy bound for an ODE solution, contractivity of a matrix step, positivity preservation, or conservation.