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

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Backward Euler is A-stable and L-stable

Statement

Backward Euler on $y^{'} = λ y$ is the scalar update $y_{n + 1} = y_{n} + z y_{n + 1}$ , where $z = h λ$ . Its stability function is

$R_{B E} (z) = \frac{1}{1 - z} .$

It is A-stable and L-stable.

Facts & Assumptions

Given: The backward-Euler scalar update, $h > 0$ , $λ \in C$ , and $z = h λ$ .

[F1]

Scalar absolute stability, A-stability, L-stability, and multistep root regions: A-stability requires $R$ to be defined and satisfy $∣ R (z) ∣ \leq 1$ for every $Re z \leq 0$ ; L-stability additionally requires the limit zero at infinity through that half-plane.

Proof

1.1givenalgebra

If $Re z \leq 0$ , then $1 - z \neq 0$ . Solving the stated update gives $y_{n + 1} = (1 - z)^{- 1} y_{n}$ , so $R_{B E} (z) = (1 - z)^{- 1}$ .

2.1F1step 1.1algebra

For $Re z \leq 0$ , $∣ 1 - z ∣^{2} - 1 = ∣ z ∣^{2} - 2 Re z \geq 0$ . Therefore $∣ R_{B E} (z) ∣ \leq 1$ on the closed left half-plane, proving A-stability by [F1].

3.1F1step 1.1step 2.1algebra∎

The reverse triangle inequality gives $∣ 1 - z ∣ \geq ∣ z ∣ - 1$ , so $∣ R_{B E} (z) ∣ \leq 1 / (∣ z ∣ - 1) \to 0$ as $∣ z ∣ \to \infty$ within $Re z \leq 0$ . By [F1] the method is L-stable.

Depends on

Scalar absolute stability, A-stability, L-stability, and multistep root regions

Used by

Dependency tree · two levels

2 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)
MIT OpenCourseWare, 18.330 Introduction to Numerical Analysis, Chapter 5, §5.2.2: Stability (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)