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M-node Gauss collocation has local defect order 2M

Statement

Assume Countable Choice for the Lebesgue interval-measure formula. Let $M \geq 1$ , fix a time $t$ and an exact IVP solution $y$ on an interval containing $[t, t + h_{0}]$ for some $h_{0} > 0$ , and put $u = y (t)$ . Let $D \subseteq R \times R^{d}$ be open and let $f : D \to R^{d}$ be an IVP vector field on a neighborhood of a common compact convex tube along this exact path, its collocation branch, and connecting segments. Assume each scalar coordinate of $f$ has continuous partial derivatives through total order $2 M + 1$ , bounded on the tube; vector smoothness and derivative estimates are componentwise, using scalar Taylor results coordinatewise. Use the $M$ -node Gaussian rule for Lebesgue measure on $[0, 1]$ , assume $∥ D_{y} f ∥_{o p} \leq L$ there, and assume the stage branch is uniquely defined in the tube for all sufficiently small $h \leq h_{0}$ . Then the exact-start local defect of the collocation endpoint is $O (h^{2 M + 1})$ as $h ↓ 0$ . Its constant depends on the fixed nodes and $M$ .

Facts & Assumptions

Given: The Gaussian moment data, its quadrature rule, the matching collocation tableau, and the exact-start IVP and local branch in the Statement.

[F1]

Positive-measure Gaussian quadrature moment data and The m-node Gaussian quadrature rule for a positive measure: for Lebesgue measure on $[0, 1]$ , the Gauss nodes and weights are defined from the monic orthogonal polynomial and cardinal integrals.

[F2]

Riemann and Lebesgue integrals agree for continuous data on [0,1]: for continuous integrands on $[0, 1]$ , the Lebesgue and Riemann integrals agree.

[F3]

A Gaussian collocation defect has a high-order weighted endpoint bound: under the stated smoothness, convex-tube and branch assumptions, the Gauss collocation endpoint defect is bounded by $C h^{2 M + 1}$ .

[F4]

Collocation tableau from distinct quadrature nodes and Collocation quadrature data are well defined: the resulting collocation coefficients and endpoint map are finite and defined.

[F5]

Lebesgue measure on the unit interval supplies Gaussian moment data: for every $M$ , Lebesgue measure on $[0, 1]$ satisfies the order- $2 M$ moment-data assumptions used in [F1].

[A1]

Countable Choice, as required by the Lebesgue moment-data verification in [F5].

[A2]

The selected stages and exact path remain in the common tube, including every segment used in the weighted-defect proof; the local branch hypotheses are uniform for sufficiently small $h$ .

Proof

1.1A1F1F4F5construct

By [F5] under [A1] the chosen Gaussian rule has admissible Lebesgue moment data. Fact [F1] gives its nodes and cardinal-integral weights, and [F4] makes the matching collocation tableau finite and defined.

2.1A2F2F3step 1.1∎

The smoothness, common-tube, and branch hypotheses of [F3] are precisely those in the Statement and [A2]. The lemma therefore gives $∥ y (t + h) - Ψ_{h} (y (t)) ∥_{2} \leq C h^{2 M + 1}$ , which is the claimed exact-start local defect estimate. Agreement of the integral conventions is [F2].

Depends on

Used by

M-node Gauss collocation has global order 2M Theorem

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Sources

Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner, Solving Ordinary Differential Equations I, 2nd ed., Chapter II §II.7, Theorems 7.9–7.10 (standard reference, not scraped)
Mathew F. Causley and David C. Seal, On the convergence of spectral deferred correction methods, Communications in Applied Mathematics and Computational Science 14 (2019), §3 (standard reference, not scraped)