= Dahlquist equivalence theorem
{c}
{wiki}
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.
Back to article page
