PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Both constant-step BDF2 test roots vanish in the stiff limit

Statement

For $Re z \leq 0$ , let $ζ$ be any root of the constant-step BDF2 test polynomial $P_{z} (ζ) = (3 - 2 z) ζ^{2} - 4 ζ + 1$ . As $∣ z ∣ \to \infty$ within the closed left half-plane, every such root satisfies $ζ \to 0$ .

This is a multistep damping statement. It does not extend the one-step definition of L-stability to multistep methods.

Facts & Assumptions

Given: A complex number $z$ with $Re z \leq 0$ and a root $ζ$ of the stated BDF2 test polynomial.

[F1]

BDF2 roots on or outside the unit disk map to the closed right half-plane: the BDF2 test polynomial on a uniform grid is $(3 - 2 z) ζ^{2} - 4 ζ + 1$ .

[F2]

Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive: complex modulus is multiplicative and satisfies the triangle inequality.

[F3]

Complex modulus restricts to real absolute value: the complex modulus of an embedded real number equals its real absolute value.

The reverse triangle inequality follows from [F2] and [F3]: $∣ u ∣ \leq ∣ u - v ∣ + ∣ v ∣$ and $∣ v ∣ \leq ∣ v - u ∣ + ∣ u ∣$ , with $∣ v - u ∣ = ∣ - 1 ∣ ∣ u - v ∣ = ∣ u - v ∣$ .

Proof

1.1givenF1F2F3algebra

Let $C = ∣ 3 - 2 z ∣$ and $r = ∣ ζ ∣$ . By [F1], the root equation implies $(3 - 2 z) ζ^{2} = 4 ζ - 1$ , hence [F2] gives $C r^{2} = ∣ 4 ζ - 1 ∣ \leq 4 r + 1$ . The reverse triangle inequality gives $C \geq ∣ ∣ 3 ∣ - ∣ 2 z ∣ ∣ = ∣ 3 - 2 ∣ z ∣ ∣ \geq 2 ∣ z ∣ - 3$ , using multiplicativity and [F3] for the real factors $2$ and $3$ . Thus $C \to \infty$ as $∣ z ∣ \to \infty$ .

2.1step 1.1choosealgebra∎

If $C > 5$ and $r \geq 1$ , step 1.1 gives $C r^{2} \leq 4 r + 1 \leq 5 r$ , hence $C \leq 5 / r \leq 5$ , a contradiction. Thus $r < 1$ for $C > 5$ , and then $C r^{2} \leq 4 r + 1 < 5$ . Given $ε > 0$ , choose $∣ z ∣$ large enough that $C > \max {5, 5 / ε^{2}}$ ; step 1.1 ensures this is possible. Then $r^{2} < 5 / C < ε^{2}$ , so $r < ε$ . This estimate holds for every root, proving the claim.

Depends on

Used by

Backward Euler and BDF2 damp a fast scalar mode while trapezoidal does not Example

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ernst Hairer and Gerhard Wanner, Introduction à l’Analyse Numérique, Chapter III, §III.9 (standard reference, not scraped)
Masayuki Yano, James Douglass Penn, George Konidaris, and Anthony T. Patera, Math, Numerics, and Programming (for Mechanical Engineers), Unit IV, Chapter 21 (standard reference, not scraped)