TheoremStatement: Literature-sourcedProof: 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.

SSPRK(3,3) preserves the degree-zero upwind entropy bound

Statement

Let $T_{h}$ be a finite periodic matching affine simplicial mesh, write $V_{K} = \int_{K} 1 d x > 0$ for its cell volumes, let $β$ be constant, and let $η : R \to R$ be convex. Assume the cellwise forward-Euler condition in The degree-zero upwind forward-Euler step preserves mass and convex entropy holds for $Δ t$ , including its Countable Choice premise through the affine-simplex Green formula. For the autonomous degree-zero upwind DG ODE, write $E_{Δ t} (U) = U + Δ t L_{h} (U)$ . The exact SSPRK(3,3) recurrence is $Y_{1} = E_{Δ t} (U^{n}), Y_{2} = \frac{3}{4} U^{n} + \frac{1}{4} E_{Δ t} (Y_{1}), U^{n + 1} = \frac{1}{3} U^{n} + \frac{2}{3} E_{Δ t} (Y_{2}) .$ Then $\sum_{K} V_{K} η (U_{K}^{n + 1}) \leq \sum_{K} V_{K} η (U_{K}^{n})$ . This guarantee is for the degree-zero scalar upwind operator just specified; it is not a fully discrete entropy theorem for the high-order nonlinear flux-differencing SBP method.

Facts & Assumptions

Given: The exact SSPRK(3,3) recurrence, the periodic degree-zero upwind spatial operator, a convex scalar entropy, and the same step size satisfying the forward-Euler bound at every stage.

[F1]

The three-stage third-order SSP Runge–Kutta step: $Y_{1} = E_{t, h} (y)$ , $Y_{2} = \frac{3}{4} y + \frac{1}{4} E_{t + h, h} (Y_{1})$ , and $y^{+} = \frac{1}{3} y + \frac{2}{3} E_{t + h / 2, h} (Y_{2})$ ; the problem here is autonomous, so all three maps are $E_{Δ t}$ .

[F2]

The degree-zero upwind forward-Euler step preserves mass and convex entropy: for every cell-state vector in $R^{T_{h}}$ , the degree-zero upwind forward-Euler map at this CFL does not increase $Φ (U) = \sum_{K} V_{K} η (U_{K})$ .

[F3]

SSPRK(3,3) is third order and preserves conditional convex bounds: if $C$ and $Φ$ are convex and the forward-Euler map preserves $C$ and does not increase $Φ$ , the SSPRK(3,3) stages and output do so at the same step size.

[F4]

A convex subset of $R^{m}$ contains every line segment between two of its points: $R^{T_{h}}$ is convex, and convexity of each $η$ with positive weights makes $Φ$ convex.

[F5]

Convex and strictly convex functions on Euclidean convex sets: $η$ is convex when $η ((1 - t) x + t y) \leq (1 - t) η (x) + t η (y)$ for $0 \leq t \leq 1$ .

[F6]

Forward-Euler step for degree-zero upwind DG transport: each cell volume $V_{K}$ is positive because every mesh cell is nondegenerate.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is carried from the forward-Euler geometric estimate [F2]; this theorem uses no stronger choice principle.

Proof

1.1givenF1F4F5F6algebra

By [F1], the spatially autonomous SSPRK(3,3) step has exactly the three displayed convex-combination stages. Set $C = R^{T_{h}}$ and $Φ (U) = \sum_{K} V_{K} η (U_{K})$ . For $U, V \in C$ and $0 \leq t \leq 1$ , apply [F5] in each cell and multiply by $V_{K} > 0$ from [F6]; summing proves convexity of $Φ$ .

2.1F2F3F4A1step 1.1

The state space $C$ is convex and $Φ$ is convex. By [F2], every forward-Euler image of a cell-state vector lies in $C$ and has no larger $Φ$ value under the assumed cell CFL. Thus the hypotheses of the convex-transfer clause [F3] hold at the same $Δ t$ .

3.1F1F3step 2.1algebra∎

Applying [F3] to the three stages in [F1] gives $Φ (U^{n + 1}) \leq Φ (U^{n})$ , which is the asserted weighted entropy inequality. The flux-differencing semidiscrete entropy theorem is not used in this transfer.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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.

Sources

Bernardo Cockburn and Chi-Wang Shu, Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems, Journal of Scientific Computing 16 (2001) (standard reference, not scraped)