= Convergence of a zero-stable multiderivative method
For a fixed-step <multiderivative multistep method>, the ordinary <zero-stability> root condition still controls propagation of the starting errors when all derivative evaluation maps are uniformly <Lipschitz continuous> on the relevant bounded region. A local defect $O(h^{p+1})$ then yields global error $O(h^p)$ over a fixed time interval, provided the starting errors are $O(h^p)$ and the implicit updates use the nearby solution branch.
For example, writing the numerical error equation as $\rho(E)e_n=hF_h(e_{n+s})+d_n$, with $F_h$ uniformly <Lipschitz continuous>, a bounded impulse response for the <root condition for a multistep method> gives
$$
\max_{j\leq n}\|e_j\|
\leq C\left(\max_{j<s}\|e_j\|+\sum_{j<n}\|d_j\|\right)
+Ch\sum_{j\leq n}\max_{i\leq j}\|e_i\|.
$$
Absorbing the last current-step term for small $h$ and applying the <discrete Gronwall inequality> gives the stated order. The higher derivatives enter through $F_h$, whose bound stays uniform as $h\to0$.
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