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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.
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A toy sequential coupling can amplify velocity error

Statement

In the scalar recurrence of Scalar added-mass staggered recurrence, set $m_{s} = 1$ , $m_{f} = 2$ , and choose $δ v_{0} \neq 0$ . Then $δ v_{n} = (- 2)^{n} δ v_{0}$ , so the increments grow geometrically in magnitude.

Facts & Assumptions

Given: The recurrence $m_{s} δ v_{n + 1} = - m_{f} δ v_{n}$ from Scalar added-mass staggered recurrence, with $m_{s} = 1$ , $m_{f} = 2$ , and $δ v_{0} \neq 0$ .

[F1]

Scalar added-mass staggered recurrence: for $m_{s}, m_{f} > 0$ , $m_{s} δ v_{n + 1} = - m_{f} δ v_{n}$ .

Verification

1.1givenF1algebra

Dividing [F1] by $m_{s} = 1$ gives $δ v_{n + 1} = - 2 δ v_{n}$ .

2.1step 1.1algebrainduction

Induction yields $δ v_{n} = (- 2)^{n} δ v_{0}$ : it holds at $n = 0$ , and multiplying the formula at $n$ by $- 2$ gives it at $n + 1$ .

3.1givenF1step 2.1algebra∎

Since $∣ δ v_{0} ∣ > 0$ , $∣ δ v_{n} ∣ = 2^{n} ∣ δ v_{0} ∣$ and $∣ δ v_{n + 1} ∣ = 2 ∣ δ v_{n} ∣ > ∣ δ v_{n} ∣$ . If instead $δ v_{0} = 0$ , step 2.1 gives zero increments throughout, so the nonzero hypothesis is needed for growth. Thus this specified recurrence is unstable in the stated scalar sense; no broader FSI claim follows.

Depends on

Scalar added-mass staggered recurrence

Used by

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Sources

Martina Maria Joosten, Aspects of interface modelling in fluid-structure interaction problems, PhD thesis, Swansea University (2011) (standard reference, not scraped)