convergence-study

name	convergence-study
description	Spatial and temporal convergence analysis with Richardson extrapolation and Grid Convergence Index (GCI) for solution verification
allowed-tools	["Bash","Read"]

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

1. Run grid/timestep refinement study with at least 3 levels
2. Compute observed convergence order with h_refinement.py or dt_refinement.py
3. Compare observed order to expected order of the scheme
4. Estimate discretization error via Richardson extrapolation
5. Report GCI for formal solution verification using gci_calculator.py
6. 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

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