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An explicit SDC sweep diverges on a stiff scalar mode

Example

Assume Countable Choice for the Lebesgue interval-measure formula. Analyze the formal one-node Gauss SDC stage-iteration recurrence for $y^{'} = λ y$ with $λ = - 6$ , $h = 1$ , and initial value $u = 1$ . The test is stiff in the sense that $h ∣ λ ∣ = 6$ ; the fully solved midpoint stage remains nonsingular and its endpoint is stable. This example studies the recurrence without invoking a residual-acceptance test.

Facts & Assumptions

Given: The one-node Gauss tableau, the explicit-left-Euler SDC preconditioner and the scalar test equation in the Example.

[F1]

One-node Gauss data are midpoint data: $Q_{11} = 1 / 2$ and $b_{1} = 1$ .

[F2]

Finite explicit-left-Euler spectral deferred-correction sweep: for one node $Q_{Δ} = 0$ .

[F3]

SDC iteration matrix for the scalar linear test equation: stage errors obey $E^{k + 1} = B (z) E^{k}$ , with $B (z) = (I - z Q_{Δ})^{- 1} z (Q - Q_{Δ})$ .

[F4]

Collocation tableau from distinct quadrature nodes: the midpoint stage solves $U = 1 + (z / 2) U$ and its endpoint is $1 + z U$ .

[A1]

Countable Choice, required to instantiate the one-node Gauss rule for Lebesgue measure on $[0, 1]$ .

Verification

1.1A1F1F2F3F4inductionalgebra

Here $z = h λ = - 6$ , so [F3] and [F1] under [A1] give $B (- 6) = (- 6) (1 / 2) = - 3$ . The collocation stage is $U^{*} = 1 - 3 U^{*}$ , hence $U^{*} = 1 / 4$ . The constant predictor is $U^{0} = 1$ , so $E^{0} = 3 / 4 \neq 0$ and $E^{k} = (- 3)^{k} (3 / 4)$ by induction; its magnitude grows by a factor $3$ at every sweep.

2.1F1F3F4algebra∎

The exactly solved endpoint is $1 - 6 (1 / 4) = - 1 / 2$ , which is also the midpoint stability factor $(1 + z / 2) / (1 - z / 2) = - 1 / 2$ multiplying the initial value. Its magnitude is below one, while the SDC contraction ratio from the scalar matrix is $∣ z ∣ / 2 = 3 > 1$ . Thus stability of the fully solved midpoint step does not imply convergence of the explicit SDC iteration, and the contraction theorem correctly makes no claim here.

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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), §3 (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)