the Lax equivalence theorem
/ lahks /
When you solve a PDE numerically, the thing you actually care about is CONVERGENCE: does the numerical answer approach the true solution as the grid is refined? Convergence is hard to check directly — you would need to know the true solution. The Lax equivalence theorem is the beautiful result that lets you sidestep that: it replaces the hard question (does it converge?) with two easy ones (is it consistent? is it stable?) that you can check separately.
The theorem states: for a well-posed linear initial-value problem and a consistent finite-difference scheme, the scheme CONVERGES if and only if it is STABLE. In slogan form, consistency plus stability equals convergence. Consistency you verify with Taylor series (the scheme tends to the PDE as h, k -> 0). Stability you verify with von Neumann analysis (errors are not amplified without bound). Neither alone suffices: a consistent-but-unstable scheme diverges, and a stable-but-inconsistent scheme converges to the wrong answer. Only together do they guarantee the numerical solution converges to the true one, and they upgrade each other — once you know consistency, stability becomes EQUIVALENT to convergence.
This is the organizing principle of the whole subject. It is why practitioners spend their effort proving stability (the genuinely subtle part) rather than attempting a direct convergence proof: by the theorem, for a consistent scheme, stability is convergence. The honest small print: it is stated for LINEAR, well-posed problems and assumes consistency holds in a suitable norm. For nonlinear PDEs (shocks, turbulence) there is no such clean equivalence, and stability concepts must be strengthened (total-variation bounds, entropy conditions). But within its linear domain, the Lax theorem is the bridge that ties together everything else in finite differences.
FTCS for the heat equation is consistent (order k + h^2). Von Neumann analysis says it is stable exactly when r = alpha*k/h^2 <= 1/2. By the Lax theorem, FTCS therefore CONVERGES if and only if you keep r <= 1/2 — no separate convergence proof needed. Violate the bound and the scheme is unstable, hence by the theorem also non-convergent.
Consistency + stability = convergence: prove the two easy parts, get the hard one free.
The equivalence is for LINEAR, well-posed problems. For nonlinear conservation laws (shocks, compressible flow) consistency plus linear stability does NOT guarantee convergence to the right (entropy-satisfying) solution; you need stronger tools like the Lax-Wendroff theorem and TVD/entropy conditions. Do not over-apply the clean linear slogan.