TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

Monotonicity, L-infinity stability, and consistency of the upwind HJB step

Statement

Assume the first-order HJB coefficients satisfy the hypotheses of Periodic first-order controlled Hamilton–Jacobi Cauchy problem. Under the CFL condition in CFL-monotone upwind update for first-order HJB equations, each one-control update is a convex combination of the current node and its one-sided neighbors plus $τ ℓ_{a}$ . Consequently the control-minimum update is order-preserving and $∥ U^{n + 1} ∥_{\infty} \leq ∥ U^{n} ∥_{\infty} + τ L .$ For a smooth periodic test function $ϕ$ with bounded derivatives through order two, the normalized local residual satisfies $\frac{ϕ (t_{n} + τ, x_{i}) - S_{h, τ} ϕ (t_{n}, \cdot)_{i}}{τ} = ϕ_{t} (x_{i}, t_{n}) + H (x_{i}, t_{n}, D ϕ (x_{i}, t_{n})) + O (τ + h_{\max} + η_{k} (P)),$ where $S_{h, τ}$ is the scheme map, $h_{\max} = \max_{j} h_{j}$ , $P$ bounds $∣ D ϕ ∣$ on the tested compact set, and $η_{k} (P) = 0$ for exact control minimization.

Facts & Assumptions

Given: The periodic upwind update, its CFL condition, bounded coefficient and running-cost bounds, and a smooth periodic test function.

[F1]

For fixed $a$ , the row is $G_{i}^{a} (V) = (1 - τ \sum_{j} \frac{∣ b_{a, j} ∣}{h_{j}}) V_{i} + τ \sum_{j} (\frac{b_{a, j}^{+}}{h_{j}} V_{i + e_{j}} + \frac{b_{a, j}^{-}}{h_{j}} V_{i - e_{j}}) + τ ℓ_{a},$ with periodic indices and $b^{+} = \max (b, 0)$ , $b^{-} = \max (- b, 0)$ ; the scheme takes the minimum of these rows (CFL-monotone upwind update for first-order HJB equations).

[F2]

For $g \in C^{2}$ with $∣ g^{''} ∣ \leq M$ , Taylor's theorem bounds a first-order increment by its linear term plus $M h^{2} / 2$ (A uniform derivative bound gives a uniform Taylor remainder bound).

[F3]

The HJB sign convention is $H (x, t, p) = - \min_{a \in A} (b_{a} (x, t) \cdot p + ℓ_{a} (x, t))$ (Periodic first-order controlled Hamilton–Jacobi Cauchy problem).

Proof

1.1F1givenalgebra

The CFL condition makes the center coefficient in [F1] nonnegative; all neighbor coefficients are nonnegative by $b_{a, j}^{\pm} \geq 0$ . Their sum is $1 - τ \sum_{j} ∣ b_{a, j} ∣ / h_{j} + τ \sum_{j} (b_{a, j}^{+} + b_{a, j}^{-}) / h_{j} = 1.$ Thus each row is a convex combination of nodal values plus $τ ℓ_{a}$ and is order-preserving. A pointwise minimum of order-preserving maps is order-preserving. Every row value lies in $[- ∥ U^{n} ∥_{\infty} - τ L, ∥ U^{n} ∥_{\infty} + τ L]$ , so the minimum has the same bound.

1.2F1F2F3givenalgebra

If $b_{a, j} \geq 0$ , the drift term in [F1] is $b_{a, j} (ϕ (x_{i} + h_{j} e_{j}) - ϕ (x_{i})) / h_{j}$ ; if $b_{a, j} < 0$ , it is $b_{a, j} (ϕ (x_{i}) - ϕ (x_{i} - h_{j} e_{j})) / h_{j}$ . In either case it equals $b_{a, j} ϕ_{x_{j}} (x_{i}) + O (∣ b_{a, j} ∣ h_{j} ∥ ϕ_{x_{j} x_{j}} ∥_{\infty})$ by [F2]. The time increment divided by $τ$ is $ϕ_{t} (x_{i}, t_{n}) + O (τ ∥ ϕ_{t t} ∥_{\infty})$ . The minimum is Lipschitz with respect to uniform perturbations of its entries, and replacing $A$ by $A_{k}$ changes the Hamiltonian by at most $η_{k} (P)$ . Combining these estimates and [F3] gives the stated residual.

2.1F1step 1.1step 1.2∎

The weights in [F1] show that the upwind step is a monotone, stable, consistent approximation with the stated error. This conclusion is local consistency only and asserts no solution-error rate.

Depends on

Used by

Convergence of the positive-interpolation and upwind first-order HJB schemes Theorem

Dependency tree · two levels

14 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

Endre Süli, B6.1 Numerical Solution of Partial Differential Equations (standard reference, not scraped)
Hung Vinh Tran, Hamilton–Jacobi Equations: Theory and Applications, 2020 author manuscript (standard reference, not scraped)