Linear multistep methods, consistency, order, and the root condition

Let $k\ge 1$, $d\ge 1$, and let $f:U\to {\mathbb{R}}^d$ be the right-hand side of a first-order initial-value problem on an open set $U\subseteq \mathbb{R}\times {\mathbb{R}}^d$, using the convention of First-order systems, initial value problems, and solutions on intervals. A constant-step $k$-step linear multistep method has real coefficients ${\alpha }_0,\dots ,{\alpha }_k,{\beta }_0,\dots ,{\beta }_k$ with ${\alpha }_k\mathrm{\ne }0$ and advances values by

${\sum }_{j=0}^k{\alpha }_j{Y}_{n+j}=h{\sum }_{j=0}^k{\beta }_{j}f(t_{n+j},Y_{n+j}),t_{n+j}=t_n+jh,h>0.$

Multiplying all coefficients by the same nonzero scalar leaves the method unchanged; one may normalize ${\alpha }_k=1$. Its characteristic polynomials are

$\rho (\zeta )={\sum }_{j=0}^k{\alpha }_j{\zeta }^j,\sigma (\zeta )={\sum }_{j=0}^k{\beta }_j{\zeta }^j\mathrm{.}$

For an exact solution $y$, the unscaled local residual at $t_n$ is

${\tau }_n(h;y)={\sum }_{j=0}^k{\alpha }_{j}y(t_n+jh)-h{\sum }_{j=0}^k{\beta }_j{y}^{\mathrm{\prime }}(t_n+jh)\mathrm{.}$

The method has order at least $p$, where $p\ge 1$ is an integer, if, for every open $U\subseteq \mathbb{R}\times {\mathbb{R}}^d$, every right-hand side $f:U\to {\mathbb{R}}^d$ as above, and every exact solution $y$ of that IVP with $p+1$ continuous derivatives, ${\tau }_n(h;y)=O(h^{p+1})$ uniformly for admitted starting times in every fixed compact interval on which the indicated derivatives are bounded. It is consistent if it has order at least one. Equivalently, with the above coefficient convention, consistency is $\rho (1)=0$ and ${\rho }^{\mathrm{\prime }}(1)=\sigma (1)$.

The root condition is that every complex root of $\rho$ has modulus at most one and every root of modulus one has algebraic multiplicity one. The method is zero-stable if for each $T>0$ there is a constant $M_T$ independent of $N$ such that every solution of the homogeneous recurrence ${\sum }_j{\alpha }_j{Y}_{n+j}=0$ on the uniform grid $h=T\mathrm{/}N$ satisfies

${\max }_{0\le n\le N}\mathrm{\parallel }{Y}_n{\mathrm{\parallel }}_{\mathrm{\infty }}\le M_T{\max }_{0\le j<k}\mathrm{\parallel }{Y}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$

Here $\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}={\max }_{0\le i<d}\mathrm{\mid }{x}_i\mathrm{\mid }$ for $x\in {\mathbb{R}}^d$. The root-condition equivalence for zero-stability is established separately.