Additive per-step errors in a Lipschitzian one-step scheme accumulate geometrically: $\eta ((1+Lh{)}^n-1)\mathrm{/}(Lh)$

Let $(X,d)$ be a metric space (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric), let $V:X\to X$ map $X$ into itself, and let $L\ge 0$ and $h>0$. Assume

$d(V(x),V(y))\le (1+Lh)d(x,y)\text{for all }x,y\in X,$

which is a Lipschitz hypothesis on $V$ with constant $1+Lh$ (Lipschitz map, $\alpha$-Hölder map for rational $0<\alpha \le 1$, and contraction). Fix $z_0\in X$, put $z_{j+1}:=V(z_j)$ for $j\in \mathbb{N}$, and let $x_0:=z_0$ together with points $x_{j+1}\in X$ satisfying

$d(x_{j+1},V(x_j))\le \eta (j\in \mathbb{N})$

for a number $\eta \ge 0$ ("computational error per step"). Then for every $n\in \mathbb{N}$:

1. if $L>0$, ${\displaystyle d(x_n,z_n)\le \eta \frac{(1+Lh{)}^n-1}{Lh}}$, and if $L=0$, $d(x_n,z_n)\le n\eta$;
2. if moreover $F:X\to \mathbb{R}$ satisfies the Lipschitz bound $\mathrm{\mid }F(x)-F(y)\mathrm{\mid }\le \mathrm{\Lambda }d(x,y)$ for all $x,y\in X$ and some $0\le \mathrm{\Lambda }<\mathrm{\infty }$, then $\mathrm{\mid }F(x_n)-F(z_n)\mathrm{\mid }\le \mathrm{\Lambda }\eta \frac{(1+Lh{)}^n-1}{Lh}(L>0),\mathrm{\mid }F(x_n)-F(z_n)\mathrm{\mid }\le \mathrm{\Lambda }n\eta (L=0);$
3. if $nh\le T$ for some $T>0$ and $L>0$, then $(1+Lh{)}^n\le e^{LT}$ and therefore $d(x_n,z_n)\le \eta \frac{e^{LT}-1}{Lh}\le \eta e^{LT}\frac{T}{h}\mathrm{.}$

• The lemma is a general accumulation statement about a one-step scheme with an external per-step perturbation; it does not model any concrete floating-point arithmetic, and $\eta$ is a hypothesis on the computed points, not a derived roundoff bound. For velocity Verlet, its application requires a metric state set $(X,d)$ on which all stages are defined, $V_h:X\to X$, computed points remaining in $X$, and a positive step $h>0$. Establish separately a numerical-map bound $d(V_h(x),V_h(y))\le (1+L_{\mathrm{m}\mathrm{a}\mathrm{p}}h)d(x,y)$ for every $x,y\in X$. Then, for $nh\le T$ and $L_{\mathrm{m}\mathrm{a}\mathrm{p}}>0$, this lemma contributes $\eta (e^{L_{\mathrm{m}\mathrm{a}\mathrm{p}}T}-1)\mathrm{/}(L_{\mathrm{m}\mathrm{a}\mathrm{p}}h)$ to the finite-time error comparison, or $n\eta$ when $L_{\mathrm{m}\mathrm{a}\mathrm{p}}=0$. The exact-field derivative constant in Finite-time global error of the velocity-Verlet method: $\parallel z_n-{\mathrm{\Phi }}_{nh}(z_0)\parallel \le C(e^{LT}-1)h^2\mathrm{/}L$ bounds the exact flow and does not establish this map bound or invariance of $X$. Combining the estimates requires their hypotheses on a common admissible set; exact-arithmetic symplecticity in The drift, kick and velocity-Verlet maps are canonical: $(DC{)}^{\mathsf{T}}JDC=J$ and $C^{\ast }\omega =\omega$ does not itself provide these error hypotheses.
• No choice principle is used; the induction is finite and the geometric sums are finite sums.