Differentiation Schemes

Differentiation Schemes

Goal

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

Requirements

• Python 3.8+
• 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:

   python3 scripts/​scheme_selector.py --smooth --periodic --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 --order 2 --accuracy 2 --scale 1.0 --json

Error Handling

Interpretation Guidance

Stencil Properties

Truncation Error Scaling

Common Stencils

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

• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 3 differentiation scripts