I am reading An Introduction to Numerical Analysis by Atkinson, and I want to make sure that I am correctly interpreting the author's definition of stability for a multistep method. In this context, a multistep method is a method of the form
$$ y_{n+1} = \sum_{j=0}^{p} a_j y_{n-j} + h \sum_{j=-1}^{p} b_j f(x_{n-j},y_{n-j}), \quad n \geq p \qquad (*) $$
for solving the initial value problem $Y'(x) = f(x,Y(x)), \; Y(x_0) = Y_0$, with $f:[x_0,b] \times \mathbb{R} \to \mathbb{R}$ assumed to be uniformly Lipschitz in its second argument. The $x_n$'s are uniformly spaced nodes in the interval $[x_0,b]$, with step size $h := x_{n+1} - x_n$. The initial values $y_0,\ldots,y_p$ are assumed to be given. On page 394, Atkinson gives the following definition of stability for $(*)$:
Let $\{y_n:0 \leq n \leq N(h)\}$ be the solution of $(*)$ for some differential equation $Y'(x) = f(x,Y(x)), \; Y(x_0) = Y_0$ for all sufficiently small values of $h$, say $h \leq h_0$. Recall that $N(h)$ denotes the largest subscript $N$ for which $x_N \leq b$. For each $h \leq h_0$, perturb the initial values $y_0,\ldots,y_p$ to new values $z_0,\ldots,z_p$ with $$ \max_{0 \leq n \leq p} |y_n - z_n| \leq \epsilon, \quad 0 < h \leq h_0. $$
Note that these initial values are likely to depend on $h$. We say the family of solutions $\{y_n: 0 \leq n \leq N(h) \}$ is stable if there is a constant $C$, independent of $h \leq h_0$ and valid for all sufficiently small $\epsilon$, for which $$ \max_{0 \leq n \leq N(h)} |y_n - z_n| \leq C \epsilon, \quad 0 < h \leq h_0. $$
Consider all differential equation problems $$ Y'(x) = f(x,Y(x)), \quad Y(x_0) = Y_0 $$ with derivative $f$ continuous and satisfying the Lipschitz condition (6.2.12) [i.e. $f$ is uniformly Lipschitz in its second argument], and suppose the approximating solutions $\{y_n\}$ are all stable. Then we say that $(*)$ is a stable numerical method.
My interpretation. (To emphasize the dependence of the iterates on the step size $h$, I'll write $y_n(h)$ and $z_n(h)$ for $y_n$ and $z_n$, respectively.)
Let $h_0 > 0$ be fixed. For each $h \in (0,h_0]$ let $\epsilon_0(h),\ldots,\epsilon_p(h) \in \mathbb{R}$ and define $z_n(h) := y_n(h) + \epsilon_n(h)$ for $n=0,1,\ldots,p$. Then define $z_{n}(h)$ for $n > p$ by the recursive equation $$ z_{n+1}(h) := \sum_{j=0}^{p} a_j z_{n-j}(h) + \sum_{j=-1}^{p} b_j f(x_{n-j}(h),z_{n-j}(h)). $$
The family of numerical solutions
$$ \mathcal{Y}(f,h_0) := \left\{ (y_n(h))_{n=0}^{N(h)}: h \in (0,h_0] \right\}$$
is called stable if there exist constants $\epsilon_0 > 0$ and $C \geq 0$ independent of $h$ such that
$$ \max_{0 \leq n \leq p} |\epsilon_n(h)| \leq \epsilon \implies \max_{0 \leq n \leq N(h)} |z_n(h) - y_n(h)| \leq C\epsilon $$
for all $\epsilon \leq \epsilon_0$ and all $h \leq h_0$.
Let $\mathcal{F}$ denote the collection of all functions $f:[x_0,b] \times \mathbb{R} \to \mathbb{R}$ which are continuous and uniformly Lipschitz in their second argument. Then $(*)$ is a stable numerical method if the family of solutions $\mathcal{Y}(f,h_0)$ is stable for each $f \in \mathcal{F}$ and each $h_0 > 0$.
Questions.
- Are my interpretations of the stability definitions correct?
- What does it mean when it says that the perturbations are "likely" to depend on $h$? Should I be thinking of the $\epsilon_n(h)$'s as round-off errors?
Edit: Let me make another attempt at stating the definition of stability, after Lutz Lehmann's feedback.
Attempt 2. The family of numerical solutions $$ \Big\{ \big((y_n(h) \big)_{n=0}^{N(h)}: h > 0 \Big\}$$ is stable if there exist constants $\epsilon_0 >0, h_0 > 0$ and $C \geq 0$ such that, if $0 < h \leq h_0$ and $0 < \epsilon \leq \epsilon_0$ and $z_0,\ldots,z_p \in \mathbb{R}$, the following implication holds: $$ \max_{0 \leq n \leq p} |y_n(h) - z_n| \leq \epsilon \implies \max_{0 \leq n \leq N(h)} |y_n(h) - z_n(h)| \leq C \epsilon. $$
Is this version correct?
@2. Of course the starter points $y_0,...,y_p$ from the exact solution will depend on $h$, as their location $x_n(h)=x_0+hn$ depends on $h$. Often the practically used starter points $z_0,...,z_p$ will be computed with a Runge-Kutta method in fixed-step mode. The deviations $\epsilon_n(h)$ thus depend on step size and method. The floating-point error is usually not considered in these theoretical explorations.
@1. You have somewhat restricted the definition, as in the definition the starter values are arbitrary in a ball around the exact values, while your interpretation restricts them to a path. Considering all paths makes the admissible set of perturbations unnecessarily larger, just considering the balls of radius $\epsilon$ appears more simple.