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Forward Euler's stability region restricts both decay rate and frequency

Facts & Assumptions

Given: $λ = - ρ + i ω$ with $ρ > 0$ , $ω \in R$ , and a step size $h > 0$ .

[F1]

Forward Euler has the closed unit disk centered at −1 as its absolute-stability region: forward Euler is absolutely stable exactly when $∣ 1 + h λ ∣ \leq 1$ .

Example

For this decaying oscillatory scalar mode, forward Euler is absolutely stable exactly when $h \leq 2 ρ / (ρ^{2} + ω^{2})$ . Thus a large oscillation frequency reduces the allowed step even when the real decay rate $ρ$ is fixed.

Verification

1.1givenalgebra

Set $z = h λ = - h ρ + i h ω$ . Then $∣ 1 + z ∣^{2} = (1 - h ρ)^{2} + (h ω)^{2} = 1 - 2 h ρ + h^{2} (ρ^{2} + ω^{2})$ .

2.1F1step 1.1algebra∎

By [F1], stability is equivalent to the last expression being at most one. Since $h > 0$ and $ρ^{2} + ω^{2} > 0$ , this is equivalent to $h (ρ^{2} + ω^{2}) \leq 2 ρ$ , or $h \leq 2 ρ / (ρ^{2} + ω^{2})$ .

Depends on

Forward Euler has the closed unit disk centered at −1 as its absolute-stability region

Used by

Nothing in the library uses this result yet.

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)