Lady Windermere's Fan¶
A telescoping identity decomposing global numerical error into locally introduced defects carried forward by later evolution.
Core Idea¶
Lady Windermere's Fan is a telescoping error identity that expresses a time-stepping method's global error as the sum of local defects propagated through subsequent exact or numerical evolution.
Replace one exact step at a time by the numerical step; adjacent mixed trajectories cancel, leaving global error. Each local defect is multiplied by later flow sensitivity before contributing at final time.
Scope of Application¶
- Numerical ODEs. Proves global error bounds.
- Time integration. Relates one-step defects to final accuracy.
- Stability analysis. Controls amplification.
- PDE discretization. Extends telescoping arguments.
Clarity¶
Include telescoping decompositions that insert and subtract mixed exact/numerical trajectories to represent global error as propagated local errors. Exclude generic triangle inequalities, local truncation error alone, and convergence claims with no stability control. Inclusion test: Include telescoping decompositions that insert and subtract mixed exact/numerical trajectories to represent global error as propagated local errors. Exclusion test: Exclude generic triangle inequalities, local truncation error alone, and convergence claims with no stability control. Nearest boundary: A simple sum of local error magnitudes is a bound only when propagation factors are controlled. Exit condition: The identity loses its meaning when exact and numerical evolution operators or common time grid are undefined. Common misclassifications: It is not local error alone. It is not a comedy reference in this context. It is not convergence without stability. It is not unweighted accumulation by default. Nearest named distinctions: Local truncation error: One source term. Global error: The final accumulated difference. Discrete Grönwall inequality: Often bounds propagation. Telescoping series: The general algebraic pattern.
Manages Complexity¶
High-order defects can still grow under unstable propagation. The telescope is algebraic while useful bounds require Lipschitz or stability data.
Abstract Reasoning¶
- Exact flow — Advances the true solution. No baseline means no error definition.
- Numerical step — Advances an approximation. Without it there is no local defect.
- Local error — Compares one numerical step with exact evolution from the same state. Global error is not one local defect.
- Insertion sequence — Replaces exact steps by numerical ones successively. Unordered subtraction does not yield the identity.
- Propagation operator — Carries each defect to final time. Unstable dynamics can magnify small defects.
- Global error — Sums all propagated contributions. Naive summing without stability is incomplete.
Knowledge Transfer¶
Telescoping local-error transport transfers across stable time-stepping schemes after exact and numerical propagators are defined; local order alone does not guarantee global accuracy.
Relationships to Other Abstractions¶
Current abstraction Lady Windermere's Fan Domain-specific
Parents (1) — more general patterns this builds on
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Lady Windermere's Fan is a kind of Decomposition Prime
Lady Windermere's Fan is a strict kind of Decomposition: it telescopically decomposes global numerical error into transported local defects.
Hierarchy path (1) — routes to 1 parentless root
- Lady Windermere's Fan → Decomposition
Neighborhood in Abstraction Space¶
Lady Windermere's Fan sits in a moderately populated region (41st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Physical & Geometric Dynamical Quantities (29 abstractions)
Nearest neighbors
- Shock-Capturing Method — 0.88
- Fourier–Bros–Iagolnitzer Transform — 0.87
- Compact Quasi-Newton Representation — 0.87
- Seismic Site Effects — 0.87
- Complex Affine Space — 0.87
Computed from structural-signature embeddings · 2026-10-08