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

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