DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Lie and Strang compositions of exact split flows

Definition

Let $a, b : U \to R^{d}$ be $C^{3}$ fields on an open set, and restrict attention to points and time intervals for which the unique local solutions of $x^{'} = a (x)$ and $x^{'} = b (x)$ remain in $U$ . Denote their exact solution maps by $Φ_{a}^{t}$ and $Φ_{b}^{t}$ . The Lie split map applies $a$ and then $b$ : $L_{h} : = Φ_{b}^{h} \circ Φ_{a}^{h} .$ The Strang split map applies half of $a$ , then $b$ , then half of $a$ : $S_{h} : = Φ_{a}^{h / 2} \circ Φ_{b}^{h} \circ Φ_{a}^{h / 2} .$ Define the vector-field Lie bracket convention $[a, b] (y) : = D b (y) a (y) - D a (y) b (y) .$ This definition is finite-dimensional. It asserts neither that an unbounded operator has a common invariant domain nor that its commutators are bounded.

Depends on

Used by

Dependency tree · two levels

6 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

Ernst Hairer, Geometric Numerical Integration, Lecture 2: Symplectic Integrators (standard reference, not scraped)
Gilbert Strang, On the Construction and Comparison of Difference Schemes (standard reference, not scraped)