eric861129/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

| Input | Description | Example | |-------|-------------|---------| | Derivative order | First, second, etc. | 1 or 2 | | Target accuracy | Order of truncation error | 2 or 4 | | Grid type | Uniform, nonuniform | uniform | | Boundary type | Periodic, Dirichlet, Neumann | periodic | | Smoothness | Smooth or discontinuous | smooth |

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

| Situation | Recommended Scheme | |-----------|-------------------| | Smooth, periodic | Central, spectral | | Smooth, bounded | Central + one-sided BCs | | Advection-dominated | Upwind | | Shocks/fronts | TVD, WENO | | High accuracy needed | Compact (Padé), spectral |

Script Outputs (JSON Fields)

| Script | Key Outputs | |--------|-------------| | scripts/stencil_generator.py | offsets, coefficients, order, accuracy | | scripts/scheme_selector.py | recommended, alternatives, notes | | scripts/truncation_error.py | error_scale, order, notes |

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

| Error | Cause | Resolution | |-------|-------|------------| | order must be positive | Invalid derivative order | Use 1, 2, 3, ... | | accuracy must be even for central | Odd accuracy requested | Use 2, 4, 6, ... | | Unknown scheme | Invalid scheme type | Use central, upwind, compact |

Interpretation Guidance

Stencil Properties

| Property | Meaning | |----------|---------| | Symmetric offsets | Central scheme (no directional bias) | | Asymmetric offsets | One-sided or upwind scheme | | More points | Higher accuracy but wider stencil |

Truncation Error Scaling

| Accuracy Order | Error Scales As | Refinement Factor | |----------------|-----------------|-------------------| | 2nd order | O(dx²) | 2× refinement → 4× error reduction | | 4th order | O(dx⁴) | 2× refinement → 16× error reduction | | 6th order | O(dx⁶) | 2× refinement → 64× error reduction |

Common Stencils

| Derivative | Accuracy | Points | Coefficients (× 1/dx or 1/dx²) | |------------|----------|--------|-------------------------------| | 1st | 2 | 3 | [-1/2, 0, 1/2] | | 1st | 4 | 5 | [1/12, -2/3, 0, 2/3, -1/12] | | 2nd | 2 | 3 | [1, -2, 1] | | 2nd | 4 | 5 | [-1/12, 4/3, -5/2, 4/3, -1/12] |

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