ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

A constant upwind DG state with inflow data on one interval

Example

Take the one-cell interval $K = (0, 1)$ , degree $p = 0$ , speed $a > 0$ , constant cell state $U (t)$ , and inflow value $g (t)$ at $x = 0$ . The outflow trace at $x = 1$ is $U (t)$ .

Facts & Assumptions

Given: The one-dimensional upwind DG weak form on $(0, 1)$ , with $β = a > 0$ , cell length one, and left inflow flux $- a g$ .

[F1]

Upwind discontinuous Galerkin discretization of constant transport: for a degree-zero test function the volume derivative term vanishes, and the two endpoint fluxes enter with their outward signs.

Verification

1.1givenF1algebra

Use the constant test $v_{h} = 1$ in the DG equation. The cell mass is $U$ , the outflow contribution is $a U$ , and the inflow contribution is $- a g$ , so $\dot{U} + a U - a g = 0$ and $\dot{U} = a (g - U)$ .

2.1givenstep 1.1algebra∎

For $E (t) = \frac{1}{2} U (t)^{2}$ , multiply the ODE by $U$ to get $\dot{E} = a U (g - U) = \frac{a}{2} g^{2} - \frac{a}{2} U^{2} - \frac{a}{2} (U - g)^{2}$ . Rearranging displays the outflow energy, mismatch dissipation at inflow, and energy supplied by the boundary datum.

Depends on

Upwind discontinuous Galerkin discretization of constant transport

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.