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

Script Outputs (JSON Fields)

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

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Do you have 3+ refinement levels?
+-- YES --> Run h_refinement.py or dt_refinement.py
|           +-- Observed order matches expected? --> Order agreement; check persistence and error assumptions
|           +-- 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

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# 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

Interpretation Guidance

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 supports the expected error law but does not establish an 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

When explaining a refinement result, compare only the dimensionless results.mean_order with the expected order. The results.richardson_extrapolated_value estimates the observable at zero spacing and has the same units as the input observable; it is never the convergence order. Use order_comparison, extrapolation_evidence, and scientific_interpretation from h_refinement.py to keep these roles explicit. For h=(0.2,0.1,0.05) and values=(1.04,1.01,1.0025), p=2 and the extrapolated observable=1. Compare 2 with the expected order, not 1. Keep this guidance in the final synthesis context. A single grid triplet cannot establish stable order across triplets: in_asymptotic_range is null for a positive single estimate. More grids, residual convergence and model validation remain needed.

Use the deterministic explanation renderer for final scientific text from trusted, input-verified h_refinement JSON. An order agreement decision is true when the measured order is within the stated tolerance, even though physical accuracy and asymptotic behavior remain unproven. The relative tolerance 0.1 means 10%, not 0.1%.

• 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.
• Checked results.in_asymptotic_range and all notes. Null means insufficient evidence; true is only a successive-order heuristic, not proof. Label any quoted extrapolation conditional on the power-law error assumption, and suppress it for invalid or diverging sequences.
• 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

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