Differentiation Schemes

Goal

Provide a reliable workflow to select a differentiation scheme, generate stencils, and assess accuracy for simulation discretization.

Requirements

• Python 3.10+
• NumPy (for stencil computations)
• No heavy dependencies

Inputs to Gather

Decision Guidance

Scheme Selection Flowchart

Is the field smooth?
├── YES → Is domain periodic?
│   ├── YES → Use central differences or spectral
│   └── NO → Use central interior + one-sided at boundaries
└── NO → Are there shocks/discontinuities?
    ├── YES → Use upwind, TVD, or WENO
    └── NO → Use central with limiters

Quick Reference

Script Outputs (JSON Fields)

Workflow

1. Identify requirements - derivative order, accuracy, smoothness
2. Select scheme - Run scripts/scheme_selector.py
3. Generate stencils - Run scripts/stencil_generator.py
4. Estimate error - Run scripts/truncation_error.py
5. Validate - Test with manufactured solutions or grid refinement

Conversational Workflow Example

User: I need to discretize a second derivative for a diffusion equation on a uniform grid. I want 4th-order accuracy.

Agent workflow:

1. Select appropriate scheme (boundary type was not stated; if the domain is bounded rather than periodic, add --boundary to surface one-sided/ghost-cell guidance, or ask the user):

   python3 scripts/scheme_selector.py --smooth --order 2 --accuracy 4 --json

2. Generate the stencil:

   python3 scripts/stencil_generator.py --order 2 --accuracy 4 --scheme central --json

3. Result: 5-point stencil with coefficients [-1/12, 4/3, -5/2, 4/3, -1/12] / dx².

Pre-Discretization Checklist

• Confirm derivative order and target accuracy
• Choose scheme appropriate to smoothness and boundaries
• Generate and inspect stencils at boundaries
• Estimate truncation error vs physics scales
• Verify with grid refinement study

CLI Examples

# Select scheme for smooth periodic problem
python3 scripts/scheme_selector.py --smooth --periodic --order 1 --accuracy 4 --json

# Generate central difference stencil for first derivative
python3 scripts/stencil_generator.py --order 1 --accuracy 2 --scheme central --json

# Generate 4th-order second derivative stencil
python3 scripts/stencil_generator.py --order 2 --accuracy 4 --scheme central --json

# Estimate truncation error
python3 scripts/truncation_error.py --dx 0.01 --accuracy 2 --scale 1.0 --json

Error Handling

Interpretation Guidance

Stencil Properties

Truncation Error Scaling

Common Stencils

Verification checklist

Before trusting a generated stencil or accepting a scheme recommendation, record concrete evidence for each item below:

• Ran stencil_generator.py --json and confirmed results.accuracy matches the requested order AND that len(results.offsets) equals the expected stencil width (e.g. 5 points for a 4th-order central second derivative); for a central scheme also verified the offsets are symmetric about 0.
• Sanity-checked the returned coefficients against references/stencil_catalog.md: confirmed they sum to ~0 (consistency: the operator annihilates a constant) and reproduce a known catalog stencil for at least one standard case (e.g. 2nd-order d²/dx² gives [1, -2, 1]/dx²).
• For a central scheme, confirmed --accuracy is even (odd values exit 2 with accuracy must be even for central); recorded the actual exit code rather than assuming the requested order was achieved.
• Recorded the error_scale, order, and reduction_if_halved from truncation_error.py and confirmed reduction_if_halved == 2**accuracy, then compared error_scale against the smallest physical feature size (dx/L_feature from references/error_guidance.md) to confirm the grid actually resolves the physics.
• Ran an independent grid-refinement / manufactured-solution study on >=3 grids (the scripts do NOT do this) and confirmed the observed order p_obs = log(e_h/e_{h/2})/log(2) is within ~10% of the formal accuracy before quoting that order.
• For a bounded (non-periodic) domain, confirmed boundary stencils were generated/selected explicitly (--scheme forward|backward or --boundary guidance) per references/boundary_handling.md, since the interior stencil alone does not define the scheme order at the boundary.
• For non-smooth fields (shocks/fronts), confirmed scheme_selector.py did NOT recommend high-order central FD and that a limiter/WENO/upwind path was chosen instead.

Common pitfalls & rationalizations

Security

Input Validation

• --order (derivative order) is validated as a positive integer with an upper bound (order <= 6)
• --accuracy is validated as a positive integer (<= 8), and additionally must be even for central schemes
• --scheme is validated against a fixed allowlist (central, forward, backward)
• --offsets (custom stencil) is length-capped (max 51), parsed as distinct integers, and must exceed the derivative order
• --dx and --scale are validated as finite, non-negative numbers (--dx strictly positive)
• No user-supplied strings are interpolated into code paths or shell commands

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

• Read: Used to inspect script source, references, and user configuration files
• Bash: Used to execute the three Python scripts (stencil_generator.py, scheme_selector.py, truncation_error.py) with explicit argument lists
• Write: Used to save generated stencil coefficients or scheme recommendations; writes are scoped to the user's working directory
• Grep/Glob: Used to locate relevant files and search references

Safety Measures

• No eval(), exec(), or dynamic code generation
• All subprocess calls use explicit argument lists (no shell=True)
• Stencil computation uses only small, bounded arrays (derivative order capped at 6, accuracy at 8, and custom offset lists capped at 51 points)
• All output is deterministic JSON with no shell-interpretable content

Limitations

• Boundary handling: Stencil generator provides interior stencils; boundaries need special treatment
• Nonuniform grids: Standard stencils assume uniform spacing
• Spectral: Not covered by stencil generator

References

• references/stencil_catalog.md - Common stencils
• references/boundary_handling.md - One-sided schemes
• references/scheme_selection.md - FD/FV/spectral comparison
• references/error_guidance.md - Truncation error scaling

Version History

• v1.2.2 (2026-06-24): Added a Verification checklist (evidence-based, tied to script JSON outputs and the references) and a Common pitfalls & rationalizations table.
• v1.2.0 (2026-06-23): Enforced even-accuracy and order upper-bound validation, corrected Security/error-handling/output docs to match scripts, fixed CLI/eval examples, hardened input validation
• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 3 differentiation scripts