the Lax equivalence theorem

Here is the question that haunts everyone who discretizes a PDE: I have built a finite-difference scheme, I have refined the grid — but does my numerical answer actually approach the true solution? The Lax equivalence theorem gives a clean, almost magical answer for linear, well-posed problems. It says you only need to check two simpler things, and convergence follows automatically.

The two things are consistency and stability. Consistency means the scheme really is an approximation of the right equation — its truncation error tends to zero as the grid is refined (so refining the grid makes the discrete equation hug the true PDE). Stability means errors do not blow up as you march forward — small perturbations stay bounded under refinement, rather than being amplified step after step until they swamp the answer. The theorem states: for a consistent finite-difference scheme of a well-posed linear initial-value problem, stability is the necessary and sufficient condition for convergence. In a slogan: consistency + stability = convergence. The deep point is that convergence (which is hard to check directly) is reduced to stability (which a Fourier/von Neumann analysis usually settles by hand).

This theorem is the theoretical backbone of the whole subject — it is why we obsess over stability conditions like the CFL limit. Read its three boxes carefully, though: it is for LINEAR, WELL-POSED problems, and it does NOT tell you the rate of convergence (consistency gives the order, stability gives boundedness, together they give that the error goes to zero, but the constant and rate need more work). For nonlinear problems — conservation laws with shocks, for instance — the clean equivalence fails and is replaced by harder results like the Lax–Wendroff theorem.