name: convergence-study description: > Perform spatial and temporal convergence analysis for solution verification — compute observed convergence orders from grid or timestep refinement studies, apply Richardson extrapolation to estimate discretization error, and calculate the Grid Convergence Index (GCI) per ASME V&V 20 standards. Use when verifying that a numerical solution converges at the expected rate, estimating the error on the finest mesh, checking whether grids are in the asymptotic range, or preparing formal verification reports, even if the user only asks "is my mesh fine enough" or "how accurate is my solution." allowed-tools:

• Bash
• Read metadata: author: HeshamFS version: "1.1.0" security_tier: high security_reviewed: true tested_with:
  • claude-code
  • gemini-cli
  • vs-code-copilot eval_cases: 2 last_reviewed: "2026-03-26"

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

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

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

Interpretation Guidance

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 greater than 1.0

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