Throughout, \(\|\cdot\|\) denotes the appropriate maximum norm. First, a lemma.

Lemma 1 (bounding the interpolation operator)

Let points \(x\in\Re^m\) and corresponding values \(U\in \Re^m\) be given, and let \(p(x)\) denote their degree \(m-1\) polynomial interpolant. Define \(h = \max_{i,j}|x_i-x_j|\) and suppose that \(\max_{i,j}|U_i-U_j| \le Ch\). Then

\[\|p(x)\| \le \|U\| + \mathcal{O}(h).\]

Now consider the iteration \[U^{n+1} = E_h U^n\] resulting from application of the CPM to \(u_t = 0\). The operator \(E_h\) forms polynomial interpolants from values of \(U^n\) and then replaces them with interpolated values. By Lemma 1, we have \(\|E_h\| \le 1 + \mathcal{O}(h)\).

Theorem 1 (Lax-Richtmeyer stability for \(u_t=0\))

Let \(T=Nk\) be fixed, and let \(U^{n+1} = E_h U^n\). Let \(h = \mathcal{O}(k)\). Then there exists a constant \(\alpha\) independent of \(k\) such that \(U^N \le e^{\alpha T}U^0\).

Proof. We have \[\|U^{n+1}\| = \|E_h U^n\| \le (1+\mathcal{O}(h))\|U^n\| = (1+ \alpha k)\|U^n\|.\] Thus \[\|U^N\| \le (1+\alpha k)^N \|U^0\| \le e^{\alpha T}U^0\]

Remarks

Unfortunately, we have to take \(h=\mathcal{O}(k)\) (i.e., \(k\ge Ch\)) which may not be reasonable. It would be much better if we could somehow show that the interpolation operator’s norm is \(1+\mathcal{O}(h^{p+1})\), but this may be difficult because the closest point operator sort of scrambles the values after interpolating…