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The trapezoidal rule is A-stable but not L-stable

Statement

The trapezoidal rule on $y^{'} = λ y$ is

$y_{n + 1} = y_{n} + \frac{z}{2} (y_{n} + y_{n + 1}), z = h λ .$

Its stability function is $R_{T} (z) = (1 + z / 2) / (1 - z / 2)$ . It is A-stable but not L-stable.

Facts & Assumptions

Given: The trapezoidal 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 $∣ R (z) ∣ \leq 1$ 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 $1 - z / 2 \neq 0$ . Solving the stated update gives $R_{T} (z) = (1 + z / 2) / (1 - z / 2)$ .

2.1F1step 1.1algebra

For $Re z \leq 0$ , $∣ 1 - z / 2 ∣^{2} - ∣ 1 + z / 2 ∣^{2} = - 2 Re z \geq 0$ . Thus $∣ R_{T} (z) ∣ \leq 1$ throughout the closed left half-plane, proving A-stability by [F1].

3.1F1step 1.1algebra∎

Along the negative real axis $z = - x$ , $x > 0$ , the multiplier is $(1 - x / 2) / (1 + x / 2)$ and tends to $- 1$ as $x \to \infty$ . Its modulus therefore tends to one, not zero, so the L-stability limit in [F1] fails.

Depends on

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

Used by

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

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)
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)