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