TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Constant-step BDF2 is A-stable under the multistep root condition

Statement

For each $z = h λ$ with $Re z \leq 0$ , the constant-step BDF2 test polynomial $P_{z} (ζ) = (3 - 2 z) ζ^{2} - 4 ζ + 1$ has degree two, all its roots have modulus at most one, and every root of modulus one is simple. Thus its multistep absolute-stability region contains the closed left half-plane, so BDF2 is A-stable.

Facts & Assumptions

Given: Constant-step BDF2 on $y^{'} = λ y$ with $h > 0$ and $Re (h λ) \leq 0$ .

[F1]

Scalar absolute stability, A-stability, L-stability, and multistep root regions: a multistep test polynomial gives absolute stability when its degree is unchanged, every root has modulus at most one, and any unit-modulus root is simple; A-stability means the closed left half-plane lies in this region.

[F2]

BDF2 roots on or outside the unit disk map to the closed right half-plane: any root of $P_{z}$ with modulus at least one forces $Re z \geq 0$ , with equality only at $z = 0, ζ = 1$ .

Proof

1.1givenF1algebra

The leading coefficient of $P_{z}$ is $3 - 2 z$ . If it vanished, then $z = 3 / 2$ , contradicting $Re z \leq 0$ . Thus $P_{z}$ has degree two.

1.2givenF2algebra

By [F2], a root with modulus greater than one would force $Re z > 0$ , which is impossible. A root with modulus one can occur only when $z = 0$ and $ζ = 1$ . At $z = 0$ , $P_{0} (ζ) = 3 ζ^{2} - 4 ζ + 1 = (3 ζ - 1) (ζ - 1)$ , so the unit root is simple and the other root is $1 / 3$ .

2.1F1step 1.1step 1.2∎

The degree and root properties in steps 1.1–1.2 satisfy the multistep criterion in [F1] for every $Re z \leq 0$ . Hence the closed left half-plane lies in the absolute-stability region, proving A-stability.

Depends on

Used by

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Dependency tree · two levels

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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)