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.

Central and upwind fluxes have different periodic dissipation

Example

On two equal periodic interval cells of length $h$ , consider degree-zero DG for $u_{t} + a u_{x} = 0$ with $a > 0$ and cell states $u_{1}, u_{2}$ . Compare the central flux ${\hat{f}}_{c} (a_{0}, b_{0}) = a (a_{0} + b_{0}) / 2$ with the upwind flux ${\hat{f}}_{u} (a_{0}, b_{0}) = a a_{0}$ .

Facts & Assumptions

Given: The degree-zero weak DG balance with a single shared numerical flux at each oriented face and periodic indexing $u_{0} = u_{2}$ , $u_{3} = u_{1}$ .

[F1]

Upwind discontinuous Galerkin discretization of constant transport: the semidiscrete cell balance is $h {\dot{u}}_{j} = - ({\hat{f}}_{j + 1 / 2} - {\hat{f}}_{j - 1 / 2})$ for a degree-zero test function.

Verification

1.1givenF1algebra

For the central flux, both faces of cell 1 see the same pair $(u_{1}, u_{2})$ and hence the same flux; likewise for cell 2. Thus ${\dot{u}}_{1} = {\dot{u}}_{2} = 0$ and $E = \frac{h}{2} (u_{1}^{2} + u_{2}^{2})$ has $\dot{E} = 0$ .

2.1F1step 1.1algebra∎

For the upwind flux, the two cell equations are ${\dot{u}}_{1} = - \frac{a}{h} (u_{1} - u_{2})$ and ${\dot{u}}_{2} = - \frac{a}{h} (u_{2} - u_{1})$ . Therefore $\dot{E} = h (u_{1} {\dot{u}}_{1} + u_{2} {\dot{u}}_{2}) = - a (u_{1} - u_{2})^{2}$ . In particular, at $(u_{1}, u_{2}) = (1, - 1)$ the central rate is zero and the upwind rate is $- 4 a < 0$ .

Depends on

Upwind discontinuous Galerkin discretization of constant transport

Used by

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Dependency tree · two levels

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