Lax equivalence theorem

Remarks

Stability implies convergence by a telescoping sum: $C(k)^n u_0 - E(nk) u_0 = \sum_{m=0}^{n-1} C(k)^{n-1-m} \left( C(k) E(mk) - E((m+1)k) \right) u_0$. Each factor $C(k)^{n-1-m}$ is bounded by $K$, and the $n$ consistency errors are each $o(k)$. Density of $D$ and the uniform bounds then cover all $u_0$. Convergence implies stability by the uniform boundedness principle, since the orbits $C(k)^n u_0$ stay bounded for each $u_0$. Example: for $u_t = u_{xx}$ on $\R$ in the maximum norm, the explicit scheme $u_j^{n+1} = u_j^n + r(u_{j+1}^n - 2u_j^n + u_{j-1}^n)$ with fixed $r = k/h^2 \le \tfrac12$ satisfies $\norm{C(k)} \le 1$, so it converges. With $r > \tfrac12$ the mode $u_j^0 = (-1)^j$ is multiplied by $1 - 4r < -1$ at each step, so the scheme is unstable and does not converge. The theorem is for linear problems; for nonlinear schemes stability and consistency do not in general characterize convergence.

Going deeper

For a student

In practice consistency is checked by Taylor expansion, and stability by a norm bound on the step operator $C(k)$, often through von Neumann's Fourier analysis in $L^2$. For explicit schemes for hyperbolic equations, stability leads to the Courant–Friedrichs–Lewy condition on $k/h$. The result is due to Lax and Richtmyer, in their 1956 paper on the stability of linear finite difference equations. The same structure, "consistency plus stability gives convergence", reappears for boundary value problems and for finite element methods, where Céa's lemma plays the role of stability.