LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Nonnegative row weights for the diagonal-diffusion HJB stencil

Statement

Under the two inequalities in Explicit monotone diagonal-diffusion Bellman grid update, every fixed-control row of its explicit update is an affine combination of the center and the two coordinate neighbors with nonnegative coefficients summing to one, plus $τ f^{a}$ . Therefore the max over controls is order-preserving.

Facts & Assumptions

Given: Positive diagonal coefficients $α_{j}^{a}$ , drift coefficients $b_{j}^{a}$ , mesh widths $h_{j}$ , and time step $τ$ satisfying the diffusion- dominance and explicit CFL conditions.

[F1]

The scheme uses $D_{j j}^{h}$ , the sign-selected $D_{j, a}^{u p}$ , and the two inequalities $∣ b_{j}^{a} ∣ h_{j} \leq α_{j}^{a}$ and $τ \sum_{j} (2 α_{j}^{a} / h_{j}^{2} + ∣ b_{j}^{a} ∣ / h_{j}) \leq 1$ (Explicit monotone diagonal-diffusion Bellman grid update).

Proof

1.1F1givenalgebra

Fix $i, a, j$ and abbreviate $α = α_{j}^{a} (x_{i}, t_{n})$ , $b = b_{j}^{a} (x_{i}, t_{n})$ , and $h = h_{j}$ . If $b \geq 0$ , the coefficient of $U_{i - e_{j}}$ in the update is $τ (α / h^{2} - b / h)$ and that of $U_{i + e_{j}}$ is $τ α / h^{2}$ . If $b < 0$ , the coefficient of $U_{i + e_{j}}$ is $τ (α / h^{2} - ∣ b ∣ / h)$ and that of $U_{i - e_{j}}$ is $τ α / h^{2}$ . Each is nonnegative by $∣ b ∣ h \leq α$ in [F1].

2.1F1step 1.1algebra

The center coefficient is $1 - τ \sum_{j} (\frac{2 α_{j}^{a}}{h_{j}^{2}} - \frac{∣ b_{j}^{a} ∣}{h_{j}}) .$ It is at least $1 - τ \sum_{j} (2 α_{j}^{a} / h_{j}^{2} + ∣ b_{j}^{a} ∣ / h_{j}) \geq 0$ by [F1]. For each coordinate, the two neighbor coefficients sum to $τ (2 α_{j}^{a} / h_{j}^{2} - ∣ b_{j}^{a} ∣ / h_{j})$ ; adding the center coefficient therefore gives one. Hence each fixed-control update is a convex combination plus its constant forcing.

3.1step 2.1algebra∎

A convex combination is order-preserving in its nodal inputs. The maximum of finitely many order-preserving control maps is order-preserving, proving the claim.

Depends on

Explicit monotone diagonal-diffusion Bellman grid update

Used by

Half-relaxed limits of the monotone diagonal-diffusion HJB scheme Theorem

Dependency tree · two levels

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Sources

Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations (standard reference, not scraped)
Guy Barles and Panagiotis E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations (standard reference, not scraped)