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 3.8+
• 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