DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Periodic one-dimensional moving grid

Definition

Fix $N \geq 1$ , a reference cell width $Δ ξ > 0$ , a time step $Δ t > 0$ , and a physical period length $L > 0$ . Face positions $x_{i + 1 / 2}^{n}$ and face velocities $w_{i + 1 / 2}^{n}$ are indexed periodically by $i \in Z$ , with $x_{i + N + 1 / 2}^{n} = x_{i + 1 / 2}^{n} + L$ and $w_{i + N + 1 / 2}^{n} = w_{i + 1 / 2}^{n}$ . The physical cell $i$ is $[x_{i - 1 / 2}^{n}, x_{i + 1 / 2}^{n}]$ and its dimensionless Jacobian is $J_{i}^{n} = (x_{i + 1 / 2}^{n} - x_{i - 1 / 2}^{n}) / Δ ξ$ ; require $J_{i}^{n} > 0$ .

For a constant physical advection speed $a \in R$ , the relative face speed is $c_{i + 1 / 2}^{n} : = a - w_{i + 1 / 2}^{n}$ . Write $c^{+} : = \max (c, 0)$ and $c^{-} : = \min (c, 0)$ . The prescribed face move is $x_{i + 1 / 2}^{n + 1} = x_{i + 1 / 2}^{n} + Δ t w_{i + 1 / 2}^{n}$ . The cell average ${\overset{ˉ}{u}}_{i}^{n}$ is the average over the corresponding physical cell.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

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)