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M-node collocation has global order M

Statement

Let $M \geq 1$ and suppose the hypotheses of M-node collocation has local defect order M and Small-step collocation maps are Lipschitz stable hold uniformly on a fixed interval $[0, T]$ . Starting with the exact initial value and using the collocation endpoint map on a uniform mesh, the maximum grid error is $O (h^{M})$ as $h = T / N ↓ 0$ .

Facts & Assumptions

Given: The exact IVP solution, a uniform mesh, the $M$ -node collocation endpoint map, and uniform tube and branch hypotheses.

[F1]

M-node collocation has local defect order M: the exact-start one-step defect is uniformly $O (h^{M + 1})$ .

[F2]

Small-step collocation maps are Lipschitz stable: for sufficiently small $h$ , the one-step map is Lipschitz with a bound $1 + L_{0} h$ for a fixed $L_{0}$ .

[F3]

Global error from uniform one-step defect and stability: on a fixed interval, exact initial data, local defect $O (h^{p + 1})$ and a $(1 + L_{0} h)$ stability bound imply global error $O (h^{p})$ .

[A1]

The exact solution, numerical grid states, stages and comparison segments stay in the common compact tube for every mesh step.

Proof

1.1A1F1F3

By [F1], take a mesh-independent constant $C$ with $∥ Ψ_{h, n} (y (t_{n})) - y (t_{n + 1}) ∥_{2} \leq C h^{M + 1}$ for every step. This is the local-defect hypothesis of [F3] with $p = M$ .

2.1A1F2F3step 1.1∎

By [F2] the same endpoint maps satisfy the fiberwise stability bound required in [F3]; the initial error is zero by hypothesis. Applying [F3] with $p = M$ proves $\max_{n} ∥ y_{n} - y (t_{n}) ∥_{2} = O (h^{M})$ .

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Sources

Ernst Hairer and Gerhard Wanner, Introduction à l’Analyse Numérique, Chapter III: Équations Différentielles Ordinaires (standard reference, not scraped)
Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner, Solving Ordinary Differential Equations I, 2nd ed., Chapter II §II.7 (standard reference, not scraped)