Convergence study

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 3.10+
• No third-party packages — scripts use only the Python standard library (math).

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, results.extrapolated_value, results.notes

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	Strongest evidence of 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	See caveat below	Confirm with order comparison
GCI asymptotic ratio far from 1.0	Not in asymptotic range	Refine further

Asymptotic-ratio caveat (constant refinement ratios). When the refinement ratios are equal (r21 == r32, the common case), the asymptotic ratio AR = GCI_coarse / (r^p * GCI_fine) reduces algebraically to f1/f2. It then only measures the relative gap between the two finest QoI values — not whether the data follow the assumed power law — so an AR near 1.0 can give false reassurance even when the observed order is far from expected. The gci_calculator.py JSON emits a notes entry flagging this. For a real asymptotic-range determination:

1. Compare the observed order p to the scheme's theoretical/expected order — a match is the meaningful evidence of being in the asymptotic range.
2. For stronger verification, use 4+ systematically refined grids and check that the observed order is consistent across successive grid triplets (h_refinement.py reports one order per triplet plus mean_order).

Verification checklist

• Used >= 3 systematically refined grids/timesteps so h_refinement.py / dt_refinement.py can report a results.mean_order; recorded the per-triplet results.observed_orders (a single-pair Richardson run does not verify order).
• Recorded results.mean_order and confirmed results.convergence_assessment reads PASS (observed order within 10% of the scheme's expected order); a FAIL or unknown means the result is not yet verified.
• Confirmed results.in_asymptotic_range is true and that no notes entry reports pre-asymptotic (>50% order variation), negative/non-positive order, or zero error differences before quoting any GCI or extrapolated value.
• Checked refinement ratios r21/r32 from gci_calculator.py are >= 1.3 (round-off noise floor) and that no Oscillatory convergence detected error was raised.
• Recorded results.gci_fine (with the safety factor used: 1.25 for >= 3 grids with verified order, 3.0 for 2 grids with assumed order) as the reported discretization uncertainty, plus results.extrapolated_value as the best estimate.
• For constant refinement ratios, did NOT treat asymptotic_ratio near 1.0 as proof of asymptotic range (it degenerates to f1/f2); confirmed the asymptotic range via observed-order-vs-expected and read the gci_calculator.py notes caveat.

Common pitfalls & rationalizations

Tempting shortcut	Why it's wrong / what to do
"Two grids agree closely, so it's converged"	Two levels cannot estimate observed order. Run h_refinement.py/dt_refinement.py with >= 3 levels; a 2-grid Richardson run uses an assumed order and needs safety factor 3.0, not 1.25.
"The asymptotic ratio is ~1.0, so we're in the asymptotic range"	With constant refinement ratios AR = GCI_coarse/(r^p*GCI_fine) reduces to f1/f2 and only measures the gap between the two finest QoI values. Verify the asymptotic range by comparing observed order to the expected order (and 4+ grid consistency).
"GCI_fine is tiny, so the solution is grid-independent"	A near-zero GCI can also mean the QoI is insensitive to refinement or the differences are in round-off noise (ratios < 1.3). Confirm in_asymptotic_range is true and refinement ratios are >= 1.3 first.
"Observed order is higher than expected, even better"	Order well above the formal order usually signals superconvergence or error cancellation, not extra accuracy. Treat it as a flag (convergence_assessment is FAIL when >10% off) and verify with more grid levels.
"The script printed an extrapolated value, so use it"	When the observed order is non-positive the solution is diverging and h_refinement.py returns richardson_extrapolated_value = null with a diverging note. Do not quote an extrapolated value or GCI in that case.
"Implicit/stable solver, so any timestep is fine for the study"	Temporal stability is not temporal accuracy. dt_refinement.py still needs >= 3 systematically reduced timesteps to recover the scheme's order; a too-coarse dt sequence stays pre-asymptotic.

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 not less than 1.0 (Roache uses Fs in {1.25, 3.0})

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