the Dahlquist equivalence theorem
/ DAHL-kvist /
Here is the question every numerical method must answer: as I shrink the step size toward zero, does my computed solution actually close in on the true one? It is not obvious that it does — small per-step errors might accumulate or amplify into nonsense. The Dahlquist equivalence theorem gives a clean, checkable answer for multistep methods: convergence holds if and only if two simpler properties both hold. It is the ODE analogue of the famous Lax equivalence theorem for PDEs.
The theorem says CONSISTENCY plus ZERO-STABILITY is equivalent to CONVERGENCE. Consistency means the method actually approximates the differential equation in the small — its local truncation error vanishes as h -> 0 (the method reproduces the true solution to at least first order; concretely the coefficients must satisfy simple sum conditions). Zero-stability means errors do not blow up under the method's own recurrence as h -> 0 — formally, the roots of the method's characteristic polynomial lie on or inside the unit circle, with any root on the circle being simple (this controls how a tiny perturbation propagates through the step-to-step recurrence). The theorem proves: a consistent method converges if and only if it is zero-stable. Neither property alone suffices — a consistent but zero-UNSTABLE method (one with a spurious root outside the unit circle) will diverge no matter how accurate each single step looks.
This is the bedrock theorem of numerical ODEs because it splits the hard global question 'does it converge?' into two local, checkable conditions. Consistency is easy to verify (Taylor-expand the local error); zero-stability is a quick root test on a polynomial. The honest subtlety: zero-stability is about the h -> 0 limit and is DIFFERENT from absolute stability (which is about a fixed nonzero h not blowing up on a decaying problem). A method can be zero-stable yet have a tiny absolute-stability region — convergent in theory but useless for stiff problems in practice. Both notions matter, and conflating them is a classic confusion.
A cautionary case: the explicit midpoint multistep formula y_{n+1} = y_{n-1} + 2h*f_n is consistent and second order, yet its characteristic polynomial has roots 1 and -1 — both on the unit circle and simple, so it IS zero-stable and converges, but the root at -1 seeds a weakly growing parasitic oscillation. Replace one coefficient so a root sits OUTSIDE the unit circle and, however consistent, the method diverges catastrophically — exactly what zero-stability forbids.
Consistency plus zero-stability if and only if convergence — the ODE Lax equivalence.
Do not confuse ZERO-stability (the h -> 0 root condition that, with consistency, gives convergence) with ABSOLUTE stability (a fixed-h boundedness on decaying problems). A method can be zero-stable and convergent yet have a useless absolute-stability region for stiff problems — both are needed, for different reasons.