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Fixed points of the SDC sweep are collocation stages

Statement

For an admissible stage vector $U$ , the fixed-point equation for one sweep of Finite explicit-left-Euler spectral deferred-correction sweep holds if and only if $U = 1 u + h Q F (U)$ , the collocation stage equation of Collocation tableau from distinct quadrature nodes.

Facts & Assumptions

Given: The nodes and collocation data, the vector field, and an admissible stage vector $U$ in the stated stage domain.

[F1]

Collocation tableau from distinct quadrature nodes: the collocation stage equation is $U = 1 u + h Q F (U)$ .

[F2]

Finite explicit-left-Euler spectral deferred-correction sweep: the sweep is defined by $U^{+} = 1 u + h {\hat{Q}}_{Δ} \hat{F} (U^{+}) + h (\hat{Q} - {\hat{Q}}_{Δ}) \hat{F} (U)$ , with $\hat{Q} = (0 ∣ Q)$ and $\hat{F} (U) = (f (t, u), F (U))$ .

Proof

1.1givenF1F2algebra

If $U$ is a fixed point of the sweep, set $U^{+} = U$ in [F2]. The two preconditioner matrices add to $\hat{Q}$ , so $U = 1 u + h \hat{Q} \hat{F} (U) = 1 u + h Q F (U)$ because the first column of $\hat{Q}$ is zero. This is [F1].

2.1givenF1F2algebra∎

Conversely, if $U = 1 u + h Q F (U)$ , then the right-hand side of [F2] with $U^{+} = U$ is $1 u + h ({\hat{Q}}_{Δ} + \hat{Q} - {\hat{Q}}_{Δ}) \hat{F} (U) = 1 u + h Q F (U) = U$ . Thus $U$ is a fixed point.

Depends on

Used by

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Sources

Mathew F. Causley and David C. Seal, On the convergence of spectral deferred correction methods, Communications in Applied Mathematics and Computational Science 14 (2019), §1.1 (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)