AlgorithmAlgorithm: AI-adaptedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Conservative upwind ALE finite-volume step

Definition

Inputs: The periodic grid and relative speeds of Periodic one-dimensional moving grid, cell averages ${\overset{ˉ}{u}}_{i}^{n} \in R$ , a constant physical speed $a$ , face velocities $w_{i + 1 / 2}^{n}$ , $Δ t > 0$ , and $Δ ξ > 0$ .

Instructions:

Without modifying the input mesh or cell averages, compute trial face positions $x_{i + 1 / 2}^{n + 1} = x_{i + 1 / 2}^{n} + Δ t w_{i + 1 / 2}^{n}$ and trial Jacobians $J_{i}^{n + 1} = (x_{i + 1 / 2}^{n + 1} - x_{i - 1 / 2}^{n + 1}) / Δ ξ$ .
If any $J_{i}^{n + 1} \leq 0$ , discard the trial data and return the status mesh_crossing and no updated state, leaving the input mesh and cell averages unchanged.
Otherwise, for every face set $c_{i + 1 / 2}^{n} = a - w_{i + 1 / 2}^{n}$ and ${\hat{F}}_{i + 1 / 2}^{n} = (c_{i + 1 / 2}^{n})^{+} {\overset{ˉ}{u}}_{i}^{n} + (c_{i + 1 / 2}^{n})^{-} {\overset{ˉ}{u}}_{i + 1}^{n}$ , with periodic indices and $c^{+} = \max (c, 0)$ , $c^{-} = \min (c, 0)$ .
Accept the trial mesh and return the status success and the unique cell averages defined by $J_{i}^{n + 1} {\overset{ˉ}{u}}_{i}^{n + 1} = J_{i}^{n} {\overset{ˉ}{u}}_{i}^{n} - (Δ t / Δ ξ) ({\hat{F}}_{i + 1 / 2}^{n} - {\hat{F}}_{i - 1 / 2}^{n})$ .

All state updates occur only after every new cell Jacobian has passed the positivity check.

Depends on

Periodic one-dimensional moving grid

Used by

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Sources

J. Donea, A. Huerta, J.-Ph. Ponthot, and A. Rodríguez-Ferran, Arbitrary Lagrangian–Eulerian Methods, Encyclopedia of Computational Mechanics, Vol. 1, Chapter 14 (2004) (standard reference, not scraped)