Dahlquist equivalence theorem

For a fixed-coefficient linear multistep method applied to a well-posed initial value problem with a continuous right-hand side uniformly Lipschitz continuous in the solution argument on the relevant region, consistency of a numerical method and zero-stability are equivalent to numerical convergence for arbitrary starting values whose errors tend to zero. The time interval is fixed while the step size tends to zero; an implicit numerical method must use its well-defined nearby solution branch for sufficiently small steps.

The quantitative statement is that a zero-stable method with consistency error $O(h^p)$ and starting errors $O(h^p)$ has global error $O(h^p)$, provided the exact solution is sufficiently smooth for that residual bound. A bounded response to step defects follows from the root condition for a multistep method, and the discrete Gronwall inequality controls the nonlinear error terms. Mere continuity of the right-hand side does not by itself supply these hypotheses or uniqueness of the exact solution.

For a linear multistep method, convergence for every set of consistent starting values requires the root condition for a multistep method. Apply the recurrence to $y^{\prime }=0$ over $N$ steps. A root $\xi$ with $|\xi |>1$ gives the error mode $e_n=|\xi {|}^{-N}{\xi }^n$: its fixed-index starting errors vanish as $N\to \infty$, but $|e_N|=1$. A unit-modulus root of multiplicity $m\ge 2$ instead gives $e_n=N^{-(m-1)}{n}^{m-1}{\xi }^n$, with the same failure. Repeated roots inside the unit disk are harmless because their polynomial growth is dominated by exponential decay.