TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Finite-time Yee error for smooth homogeneous periodic Maxwell fields

Statement

Assume the hypotheses and residual bounds of Second-order consistency of the homogeneous periodic Yee scheme on $[0, T]$ . Write $δ_{t t} u^{n} = (u^{n + 1} - 2 u^{n} + u^{n - 1}) / Δ t^{2}$ . Let $e_{h}^{n}, b_{h}^{n + 1 / 2}$ be a lossless, source-free periodic Yee solution, with $S = 0$ and $j^{n + 1 / 2} = 0$ at every step, and let $e_{*}^{n}, b_{*}^{n + 1 / 2}$ be the exact samples. Set $w^{n} = e_{h}^{n} - e_{*}^{n}$ and $z^{n + 1 / 2} = b_{h}^{n + 1 / 2} - b_{*}^{n + 1 / 2}$ . Suppose the initial electric rows are compatible with the Yee equations and satisfy $∥ w^{0} ∥_{M_{e}} + ∥ C w^{0} ∥_{M_{b}} + ∥ w^{1} ∥_{M_{e}} + ∥ C w^{1} ∥_{M_{b}} + {∥ \frac{w^{1} - w^{0}}{Δ t} ∥}_{M_{e}} + ∥ z^{1 / 2} ∥_{M_{b}} \leq C_{0} (h^{2} + Δ t^{2}) .$ Let $λ_{\max}$ be the largest generalized curl-curl eigenvalue in Strict spectral CFL positivity and Yee modified energy. If the strict spectral margin obeys $γ : = 1 - Δ t^{2} λ_{\max} / 4 \geq γ_{0} > 0$ , then for every integer $n \geq 0$ with $t_{n + 1} \leq T$ , $∥ e_{h}^{n} - e_{*}^{n} ∥_{M_{e}} + ∥ b_{h}^{n + 1 / 2} - b_{*}^{n + 1 / 2} ∥_{M_{b}} + ∥ C (e_{h}^{n} - e_{*}^{n}) ∥_{M_{b}} \leq C (h^{2} + Δ t^{2}),$ where $C$ is independent of the grid counts and time step. The statement is for the smooth homogeneous periodic branch and compatible starts; it makes no Taylor error claim across material jumps or an unspecified PEC boundary stencil.

Facts & Assumptions

Given: A homogeneous periodic grid, exact smooth source-free Maxwell samples, the lossless source-free Yee rows, a strict CFL margin, and the stated initial-error bound.

[F1]

With $S = 0$ and $j^{n + 1 / 2} = 0$ at every step, eliminating the magnetic half-step from the Yee rows gives the electric recurrence $M_{e} δ_{t t} e^{n} + C^{T} M_{b} C e^{n} = 0$ and the Faraday half-step row. (Lossless or midpoint-conductive Yee leapfrog update)

[F2]

The generalized Rayleigh maximum $λ_{\max}$ gives $∥ C v ∥_{M_{b}}^{2} \leq λ_{\max} ∥ v ∥_{M_{e}}^{2}$ for every electric cochain $v$ . The strict spectral margin is $1 - Δ t^{2} λ_{\max} / 4$ . (Strict spectral CFL positivity and Yee modified energy)

[F3]

Exact samples satisfy $∥ r_{h}^{n} ∥_{M_{e}^{- 1}} \leq C_{T} (h^{2} + Δ t^{2})$ and the unscaled Faraday-row residual satisfies $∥ s_{h}^{n} ∥_{M_{b}} \leq C_{T} Δ t^{3}$ . (Second-order consistency of the homogeneous periodic Yee scheme)

[F4]

$M_{e}$ is positive definite, and $∥ r ∥_{M_{e}^{- 1}}^{2} = r^{T} M_{e}^{- 1} r$ is the dual norm to $∥ \cdot ∥_{M_{e}}$ . (Oriented Maxwell cochains, masses, and the Yee grid)

Proof

1.1F1F3algebra

Subtract the exact-sample equations with residuals from the numerical rows. With $K = C^{T} M_{b} C$ , this gives $M_{e} δ_{t t} w^{n} + K w^{n} = - r_{h}^{n}, z^{n + 1 / 2} - z^{n - 1 / 2} = - Δ t C w^{n} - s_{h}^{n} .$ The sign convention for $s_{h}^{n}$ is the one in the consistency statement.

1.2F2givenalgebra

Put $V^{n} = (w^{n + 1} - w^{n}) / Δ t$ , ${\overset{ˉ}{w}}^{n} = (w^{n + 1} + w^{n}) / 2$ , and $E^{n + 1 / 2} = \frac{1}{2} ∥ V^{n} ∥_{M_{e}}^{2} + \frac{1}{2} ⟨ C w^{n + 1}, C w^{n} ⟩_{M_{b}} .$ Expanding about $C {\overset{ˉ}{w}}^{n}$ gives $E^{n + 1 / 2} = \frac{1}{2} ∥ V^{n} ∥_{M_{e}}^{2} + \frac{1}{2} ∥ C {\overset{ˉ}{w}}^{n} ∥_{M_{b}}^{2} - \frac{Δ t^{2}}{8} ∥ C V^{n} ∥_{M_{b}}^{2}$ . The generalized Rayleigh bound in [F2] yields $E^{n + 1 / 2} \geq \frac{γ_{0}}{2} ∥ V^{n} ∥_{M_{e}}^{2} + \frac{1}{2} ∥ C {\overset{ˉ}{w}}^{n} ∥_{M_{b}}^{2} .$ The initial assumptions therefore give $E^{1 / 2} \geq 0$ and $\sqrt{E^{1 / 2}} \leq C (h^{2} + Δ t^{2})$ .

2.1F4step 1.1algebra

The kinetic-energy increment is $Δ t ⟨ (V^{n} + V^{n - 1}) / 2, M_{e} δ_{t t} w^{n} ⟩$ , and the curl-potential increment is $Δ t ⟨ (V^{n} + V^{n - 1}) / 2, K w^{n} ⟩$ . Substituting step 1.1 gives $E^{n + 1 / 2} - E^{n - 1 / 2} = - Δ t ⟨ \frac{V^{n} + V^{n - 1}}{2}, r_{h}^{n} ⟩ .$ By [F4] and Cauchy–Schwarz, the absolute value is at most $Δ t ∥ r_{h}^{n} ∥_{M_{e}^{- 1}} (∥ V^{n} ∥_{M_{e}} + ∥ V^{n - 1} ∥_{M_{e}}) / 2$ .

3.1F2F3step 1.2step 2.1algebra

Coercivity in step 1.2 bounds both velocities by the square roots of their adjacent energies times $\sqrt{2 / γ_{0}}$ . Factoring the difference of squares and summing (with the zero-energy case obtained by continuity, or by first replacing each square root by $\sqrt{E + δ}$ and letting $δ ↓ 0$ ) gives $\max_{n} (∥ V^{n} ∥_{M_{e}} + ∥ C {\overset{ˉ}{w}}^{n} ∥_{M_{b}}) \leq C (\sqrt{E^{1 / 2}} + \sum_{n} Δ t ∥ r_{h}^{n} ∥_{M_{e}^{- 1}}) \leq C_{T} (h^{2} + Δ t^{2}) .$ The sum has at most $T / Δ t$ terms, so [F3] gives the final inequality.

4.1F2givenstep 1.2step 3.1algebra

Since $w^{n} = w^{0} + Δ t \sum_{m < n} V^{m}$ , the initial bound and step 3.1 imply $\max_{n} ∥ w^{n} ∥_{M_{e}} \leq C_{T} (h^{2} + Δ t^{2})$ . For $n \geq 1$ , $C w^{n} = C {\overset{ˉ}{w}}^{n - 1} + Δ t C V^{n - 1} / 2$ and $Δ t ∥ C V^{n - 1} ∥_{M_{b}} \leq Δ t \sqrt{λ_{\max}} ∥ V^{n - 1} ∥_{M_{e}} \leq 2 ∥ V^{n - 1} ∥_{M_{e}}$ by the strict CFL condition. Thus the same order holds for $∥ C w^{n} ∥_{M_{b}}$ for $n \geq 1$ ; the $n = 0$ curl bound is part of the initial assumptions.

5.1F3givenstep 1.1step 4.1algebra∎

Summing the second equation in step 1.1 from the controlled start $z^{1 / 2}$ gives $∥ z^{n + 1 / 2} ∥_{M_{b}} \leq ∥ z^{1 / 2} ∥_{M_{b}} + Δ t \sum_{m = 1}^{n} ∥ C w^{m} ∥_{M_{b}} + \sum_{m = 1}^{n} ∥ s_{h}^{m} ∥_{M_{b}} .$ The curl sum is bounded by $T \max_{m} ∥ C w^{m} ∥_{M_{b}}$ , and the residual sum is at most $(T / Δ t) C_{T} Δ t^{3} = C_{T} T Δ t^{2}$ by [F3]. The initial bound and step 4.1 therefore bound both terms by $C_{T} (h^{2} + Δ t^{2})$ . Combining the electric, curl and magnetic bounds proves the claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

John B. Schneider, Understanding the Finite-Difference Time-Domain Method (standard reference, not scraped)