LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Bounded per-step state perturbations add a separate linear energy term

Statement

Use the analytic data and small step of Analytic Verlet retains order-two energy accuracy on exponential times. Suppose computed points $x_{j} \in K$ satisfy $∥ x_{j + 1} - Ψ_{h} (x_{j}) ∥ \leq η$ , for $0 \leq j < n$ , with $η \geq 0$ small enough that all comparison segments lie in $D (ρ / 8)$ . Then

$∣ H (x_{n}) - H (x_{0}) ∣ \leq 2 C_{E} h^{2} + C n h e^{- c / h} + C_{G} n η .$

In particular the state-error contribution is $C_{G} (t / h) η$ for $t = n h$ . This is a deterministic additive-error hypothesis, not an analysis of floating-point arithmetic or a claim of rounded symplecticity.

Facts & Assumptions

Given: The stated computed points, perturbation bound, domain segments and analytic constants.

[F1]

The exponential local state defect and bounded gradient follow from Optimal finite truncation gives an exponential local splitting defect and The optimal modified Hamiltonian differs from the original by order two; the modified flow conserves $H_{h}$ as proved in Analytic Verlet retains order-two energy accuracy on exponential times.

Proof

1.1givenF1algebra

Per-step estimate. Compare $x_{j + 1}$ first with $Ψ_{h} (x_{j})$ and then with the modified-flow endpoint. Integrating the gradient along the two stated segments gives $∣ H_{h} (x_{j + 1}) - H_{h} (x_{j}) ∣ \leq C_{G} η + C h e^{- c / h}$ , using conservation at the modified endpoint.

2.1step 1.1F1algebra∎

Sum and compare. Summation gives $∣ H_{h} (x_{n}) - H_{h} (x_{0}) ∣ \leq C_{G} n η + C n h e^{- c / h}$ . Adding the two endpoint bounds $∣ H_{h} - H ∣ \leq C_{E} h^{2}$ proves the displayed result, including $n = 0$ and $η = 0$ .

Depends on

Dependency tree · 2 levels

Sources

Hairer, Geometric Numerical Integration, Lecture 3: backward error analysis, Sections 1–3 and 5; local analytic estimates supplied here