Lax Equivalence Theorem

The Lax equivalence theorem states that for a well-posed initial value problem and a consistent numerical method:

$$\text{stable}⟺\text{convergent}$$

In full: a consistent method is convergent if and only if it is stable.

Why It Is Useful

Convergence is what we ultimately care about, but it is often difficult to prove directly. Stability and consistency are each easier to verify:

• Consistency: check that the Local truncation error vanishes as $dt\to 0$ (typically by Taylor expanding the method).
• Stability: check that the amplification factor satisfies $|A|\le 1$.

The Lax equivalence theorem lets us prove convergence by establishing these two properties separately, rather than estimating the global error directly.

Note: the theorem applies to well-posed problems, meaning the underlying ODE must itself be stable (solutions don’t blow up from small perturbations in initial conditions).

Convergence | Stability of a method | Stability of an ODE | Local truncation error | Order of convergence