convergence-study

HeshamFS

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关于

This skill performs automated convergence analysis for numerical simulations, computing observed orders of accuracy and estimating discretization errors using Richardson extrapolation and Grid Convergence Index (GCI) calculations. It helps developers verify mesh adequacy, check asymptotic convergence, and prepare formal verification reports per ASME V&V 20 standards. Use it whenever you need to assess solution accuracy or determine if your grid resolution is sufficient.

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技能文档

Convergence Study

Goal

Provide script-driven convergence analysis for verifying that numerical solutions converge at the expected rate as the mesh or timestep is refined.

Requirements

Python 3.8+
NumPy (not required; scripts use only math stdlib)

Inputs to Gather

Input	Description	Example
Grid spacings	Sequence of mesh sizes (coarse to fine)	`0.4,0.2,0.1,0.05`
Timestep sizes	Sequence of dt values	`0.04,0.02,0.01`
Solution values	QoI at each refinement level	`1.16,1.04,1.01,1.0025`
Expected order	Formal order of the numerical scheme	`2.0`
Safety factor	GCI safety factor (1.25 default)	`1.25`

Script Outputs (JSON Fields)

Script	Key Outputs
`scripts/h_refinement.py`	`results.observed_orders`, `results.mean_order`, `results.richardson_extrapolated_value`, `results.convergence_assessment`
`scripts/dt_refinement.py`	Same as h_refinement but for temporal convergence
`scripts/richardson_extrapolation.py`	`results.extrapolated_value`, `results.error_estimate`, `results.observed_order`
`scripts/gci_calculator.py`	`results.observed_order`, `results.gci_fine`, `results.gci_coarse`, `results.asymptotic_ratio`, `results.in_asymptotic_range`

Workflow

Run grid/timestep refinement study with at least 3 levels
Compute observed convergence order with h_refinement.py or dt_refinement.py
Compare observed order to expected order of the scheme
Estimate discretization error via Richardson extrapolation
Report GCI for formal solution verification using gci_calculator.py
Document convergence results and any anomalies

Decision Guidance

Do you have 3+ refinement levels?
+-- YES --> Run h_refinement.py or dt_refinement.py
|           +-- Observed order matches expected? --> Solution verified
|           +-- Order too low? --> Check: pre-asymptotic, coding error, insufficient resolution
|           +-- Order too high? --> Check: superconvergence or cancellation effects
+-- NO (only 2 levels) --> Use richardson_extrapolation.py with assumed order
                           (less reliable without order verification)

CLI Examples

# Spatial convergence with 4 grid levels
python3 scripts/h_refinement.py --spacings 0.4,0.2,0.1,0.05 --values 1.16,1.04,1.01,1.0025 --expected-order 2.0 --json

# Temporal convergence with 3 timestep levels
python3 scripts/dt_refinement.py --timesteps 0.04,0.02,0.01 --values 2.12,2.03,2.0075 --expected-order 2.0 --json

# Richardson extrapolation with assumed 2nd-order
python3 scripts/richardson_extrapolation.py --spacings 0.02,0.01 --values 1.0032,1.0008 --order 2.0 --json

# GCI for 3-mesh verification
python3 scripts/gci_calculator.py --spacings 0.04,0.02,0.01 --values 1.0128,1.0032,1.0008 --json

Error Handling

Error	Cause	Resolution
`spacings and values must have the same length`	Mismatched input arrays	Provide equal-length lists
`At least 2 refinement levels required`	Too few data points	Add more refinement levels
`Exactly 3 refinement levels required`	GCI needs 3 levels	Provide fine/medium/coarse
`Oscillatory convergence detected`	Non-monotone convergence	Check mesh quality or scheme

Interpretation Guidance

Scenario	Meaning	Action
Observed order matches expected	Solution in asymptotic range	Report GCI, extrapolate
Observed order < expected	Pre-asymptotic or coding bug	Refine further or debug
Negative observed order	Solution diverging	Check implementation
GCI asymptotic ratio near 1.0	Grids in asymptotic range	Results are reliable
GCI asymptotic ratio far from 1.0	Not in asymptotic range	Refine further

Security

Input Validation

All numeric parameters (spacings, timesteps, values, expected-order, order) are validated as finite positive numbers
Comma-separated value lists are length-matched (spacings and values must have equal length) and capped at 10,000 entries
GCI calculator enforces exactly 3 refinement levels; Richardson extrapolation requires at least 2
Safety factor is validated as a finite number greater than 1.0

File Access

Scripts read no external files; all inputs are provided via CLI arguments
Scripts write only to stdout (JSON output); no files are created unless the agent explicitly uses the Write tool

Tool Restrictions

Bash: Used to execute the four Python analysis scripts (h_refinement.py, dt_refinement.py, richardson_extrapolation.py, gci_calculator.py) with explicit argument lists
Read: Used to inspect script source and reference documentation

Safety Measures

No eval(), exec(), or dynamic code generation
All subprocess calls use explicit argument lists (no shell=True)
Scripts use only Python standard library (math module); no pickle loading or deserialization of untrusted data
Minimal tool surface (Bash and Read only) limits the agent's ability to modify the filesystem

References

references/convergence_theory.md - Formal convergence order, log-log analysis, asymptotic range
references/gci_guidelines.md - Roache's GCI method, ASME V&V 20, safety factors

GitHub 仓库

HeshamFS/materials-simulation-skills

路径: skills/core-numerical/convergence-study

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